Magnus Academy (Maths Specialist): May 2020

Sunday, 31 May 2020

RANK CORRELATION (C. A)

RANK CORRELATION COEFFICIENT

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1) The coefficient of rank correlation of the marks obtained by 10 students in Mathematics and Statistics was found to be 0.5. It was then detected that the difference in ranks in the two subjects for one particular student was wrongly taken to be 3 in place

of 7. What should be the correct rank correlation coefficient.

A) 0.26. B) -0.26. C) 0.62 D) 0.623

2) R': 1 2 3 4 5

R": 5 4 3 2 1 find R.

A) 1. B) 0. C) - 1. D) none

3) If sum of squares of difference in two ranks is 33 and Number of variables are 10, Find the value of coefficient of rank correlation.

A) 0.8. B) -0.80. C) 0.94. D) -0.94

4) Find rank correlation coefficient

R' : 1 2 3 4 5

R":. 1 2 3 4 5

A) 1. B) 0. C) -1. D) none

5) If n=10, and ∑D² =280 Find R.

A) 0.70. B) -0.7. C) 0.645. D) none

6) If n=10 and ∑D² =30 find R.

A) 0.28. B) -0.82. C) 0.82. D) -0.28

7) If rank correlation coefficient is 0.60 and N=10 find ∑D² where D is the difference in ranks of the

two series.

A) 61. B) 64. C) 74. D) 66

8) The coefficient of rank correlation between the marks in Statistics and Mathematics obtained by a certain group of students is ⅔ and the sum of the squares of the difference in ranks is 55. How many students are there in the group ?

A) 10. B) 9.C) 12. D) more than 15

9) Find rank correlation coefficient

Roll no. Marks in. Marks in

Maths. Statistics

1 78 84

2 36 51

3 97 91

4 25 60

5 75 68

6 82 62

7 90 86

8 62 58

9 65 53

10 39 47

A) 0.28. B) -0.39. C) 0.49. D) 0.82

10) Find R

Roll no. Marks in Marks in

English. Maths

1 43 36

2 29 6

3 35 17

4 18 14

5 40 25

6 11 10

7 49 32

8 10 0

9 5 3

10 22 20

A) 0.95. B) -0.95. C) 0.82. D) 0.45

11) Find R

Roll no. Marks in Marks in

English. Maths

1 80 85

2 38 50

3 95 92

4 30 58

5 74 70

6 84 65

7 91 88

8 60 56

9 66 52

10 40 46

A) 0.87. B) 0.57. C) -0.57. D)0.49

12) find R

X: 80 91 99 71 61 81 70 59

Y:123 135 154 110 105 134 121106

A) -0.95. B) 0.785. C) 0.95 d) none

13) Find R

X: 75 88 95 70 60 80 81 50

Y:120 134 150 115 110 140 142100

A) 0.93. B) -0.83. C) 0.85. D) 0.63

14) The ranking to individuals at the start and on the finish of a course of training are given below. What is the value of Spearman's coefficient of Correlation?

Roll no: 1 2 3 4 5 6 7 8 9 10

Rank 1: 1 6 3 9 5 2 7 10 8 4

RANK2: 6 8 3 7 2 1 5 9 4 10

A) 0.947 B) - 0.489

C) 0.489. D) 0.394

15) In a contest, two judges ranked eight candidates in order of their performances, as shown in the table given below. The rank Correlation coefficient is :

Candidates: A B C D E F G H

JUDGE 1: 5 2 8 1 4 6 3 7

Judge 2: 4 5 7 3 2 8 1 6

A) - 0.678. B) 0.875

C) 0.67. D) none

16) Find R

R. N: 1 2 3 4 5 6 7 8

X: 62 53 51 25 79 43 60 33

Y:. 52 63 45 36 72 65 45 25

A) 0.489 b) -0.64 c) 0.64 d) none

17) Find the Rank correlation

X: 70 65 71 62 58 69 78 64

X: 91 76 65 83 90 64 55 48

A) 0.3125. B) - 0.3095

C) 0.2955. D) - 0.2955

18) Find Rank correlation

Roll no. Marks in. Marks in

Account. Statistics

1 30 15

2 20 40

3 40 40

4 50 45

5 30 20

6 20 30

7 30 15

8 50 50

9 10 20

10 0 10

A) 0.63. B) 0.83. C) 0.36. D) none

19) Ten competition in a beauty contest are ranked by three judges in the following order. Use rank Correlation coefficient to determine which pair of judges has the nearest approach to common tastes in beauty.

Judge 1: 1 6 5 10 3 2 4 9 7 8

Judge 2: 3 5 8 4 7 10 2 1 6 9

Judge 3: 6 4 9 8 1 2 3 10 5 7

A) 1st and 2nd judge

B) 2nd and 3rd judg6

C) 1st and 3rd judge

D) none

20) The marks secured by a group of 10 students in Written Selection Test (X) and in the Aptitude Test (Y) are given in the following table. Calculate product-moment Correlation coefficient (r) and rank Correlation coefficient (R). The value of absolute difference between "r" and "R" is :

Test (X). Test (Y)

44 24

42 25

40 28

52 29

39 32

32 35

24 36

46 41

41 45

50 50

A) 0.063 b) 0.897 c)0.01 d) 0

21) Rank correlation is useful when we study the relation between___ characteristics.
A) quantitative. B) qualitative
C) both A and B C) either A or B

22) Which of the following method (s) would be considered for finding Correlation between two attributes?
A) Pearson's product moment method.
B) Coefficient of concurrent Deviation
C) Spearman's Rank correlation coefficient
E) Both A and C.

23) Spearman's Rank correlation formula is given by
R= 1 - 6 ∑D²/(n³-n), where D stand for:
A) Sum of the rank of the two variables.
B) Difference between the rank of two variables
C) both A and B
D) either A or B

24) Spearman's Rank correlation formula 1- 6{∑D² + (m³-m))12}/(n³-n), is used when:
A) There are repeated rank in only one series
B) there are repeated ranks in any one of the series
C) There are repeated ranks in any one of the series or both of the series.
D) it is not used for repeated ranks.

25) The coefficient of rank Correlation of the marks obtained by 10 students in Mathematics and Statistics was found to be 0.5. it was then detected that the difference in ranks in the two subjects for one particular students was wrongly taken to be 3 in place of 7. What should be the correct rank Correlation coefficient?
A) +0.26 B) -0.26 C) 0.62 D) 0.623

26) If the sum of squares of difference in two ranks is 33 and number of variables are 10, find the value of Rank correlation coefficient.
A) 0.8 B) -0.8 C) 0.94 D) -0.94

27) If n=10 and ∑D² = 280, then which of the following represents the value of rank Correlation coefficient?
A) 0.7. B) -0.7. C) 0.645. D) none

28) For two series we have, ∑D²= 30 and n= 10, find the value of R
A) 0.28. B) -0.82. C) 0.82. D) -0.28

29) If R= 0.60 and n= 10. Find the value of ∑D². Where D is the difference in ranks of the two series.
A) 61. B) 64. C) 74. D) 66

30) The coefficient of rank Correlation between the marks in Statistics and Mathematics obtained by a certain group of students is 2/3 and the sum of the squares of the differences in ranks is 55. How many students are there in the group?
A) 10. B) 9.
C) 12. D) more than 15

1) a 2) c 3) a 4) a 5) b 6) c 7) d 8) a 9) d 10) c 11) a 12) c 13) a 14) d 15) c 16) c 17) b 18) a 19) c 20) a 21) b 22) c 23) b 24) c 25) a 26) a 27) b 28) c 29) d 30) a

Wednesday, 27 May 2020

CORRELATION COEFFICIENT (C)

CORRELATION COEFFICIENT

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1) Correlation between x and y is 0.52, their covariance is 7.8. If the variance of x is 16, then S.D of y is: A 2.85. B) 3.25. C) 1.25. D) 3.75

2) Find r if ∑x=50, ∑y=100, ∑x²=255, ∑y²=4594, ∑xy=630,n=10

A) 0.8678. B) 0.9698 C) 0. 9688. D) None

3) Find r if ∑x=125, ∑y=100, ∑x²= 650, ∑y²=436, ∑xy=520,n=25 A) 0.667. B) - 0.667

C) 1.667. C) 0.0667

4) Find r, n=8 ∑x=56, ∑y=40, ∑x²=524, ∑y²= 256, ∑xy= 364.

5) Find r if n=10, ∑x=140,∑y=150,

∑(x-10)²=180, ∑=(y-15)²=215,

∑(x-10)(y-15)=60

a) 0.81 b) 0.91 c) -0.81 d) -0.91

6) find r if n=12, ∑x=120, ∑y=130,

∑(x-8)²=150, ∑(y-10)²= 200,

∑(x-8)(y-10)= 50

a) 0.215 b) -0.215 c) -0.317 d) n

7) Find r if

¹²ᵢ₌₁∑(x - mean of x)²=120,

¹²ᵢ₌₁∑(y- mean of y)²=250 ,

¹²ᵢ₌₁∑(x-mean of x)(y-mn of y) = 225.

a) 0.987 b) -0.75 c) 0.75 d) 0.85

8) If cov(x,y)=12, r=0.6 and s.d of

x=5 find s.d of y.

a) 2 b) 3 c) 1 d) 4

9) If r=0.5, ∑uv=120,∑u²=90 and σᵥ=8 find n

a) 8 b) 9 c) 10 d) 12

10) cov (u,v)=3, σ²ᵤ= 4.5, σ²ᵥ=5.5, find r.

a) +0.503 b) +0.603

c) -0.603 d) -0.556

11) Find the probable error of r if

r=0.05 and n=25.

a) 0.1928 b) 0.1292

c) 0.0129 d) 0.0192

12) If r= +0.526 and n=50, what is the probable of r.

a) 0.06623 b) 0.6623

c) 1.6623 d) - 0.6629

13) ¹⁰⁰ᵢ₌₁∑x=280, ¹⁰⁰ᵢ₌₁∑y=60,

¹⁰⁰ᵢ₌₁∑x²=2384, ¹⁰⁰ᵢ₌₁∑y²=117,

¹⁰⁰ᵢ₌₁∑xy=438. find

i) Correlation coefficient

a) 0.45 b) 0.75 c) 0.95 d) 1

ii) Coefficient of determination.

a) 0.0456 b) 0.9556

c) 0.5625 d) 0.2595

iii) The percentage of variation unaccounted for is

a) 43.85% b) 56.25%

c) 65.52% d) none

iv) standard error of r is

a) 0.04234 b) 0.05784

c) 0.03585 c) 0.04375

v) Probable error of r is.

a) 0.02951 b) 0.29515

c) 0.95411 d) none

14) X: 1 2 3 4 5

Y: 3 2 5 4 6 find r.

a) 0.45 b) 0.8 c) 0.95 d) 1.00

15) X: 1 2 3 4 5.

Y: 6 8 11 8 12 Find r.

a) +0.775 b) - 0.775

c) + 0.895 d) + 0.956

16) X: 10 12 13 16 17 20 25

Y: 19 22 24 27 29 33 37

find correlation Coefficient.

a) -0.987 b) 0.99 c)0.895 d) -0.99

17) You are given the following data relating to aptitude scores(x) and productivity index(y).

x- 9 18 18 20 20 23

y- 33 23 33 42 29 32

Find the co-efficient of correlation between aptitude and productivity index.

a) 0.3333 b) 0.0237

c) 0.0337 d) 0.0733

18) Find correlation coefficient

X- 4 6 8 10 12

Y- 3 6 9 12 15

a) +1 b) - 1 c) 0 d) +0.878

19) Find r taking 44 and 26 as the

origin of X and Y

X Y

43 29

44 31

46 19

40 18

44 19

42 27

45 27

42 29

38 41

40 30

42 26

57 10

a) -0.73 b) +0.73

c) -0.537 d) +0.537

20) Calculate correlation between age and playing habits of students from the data given .

age(yrs) no. of regular

students players

15 250 200

16 200 150

17 150 90

18 120 48

19 100 30

20 80 12

a) - 0.0963 b) + 0.096

c) - 0.1963 d) + 0.0963

21) What can be said about the coefficient of correlation .

X 1 2 3 4 5 6 7

Y 3 5 6 8 10 11 13

a) r is greater than 0

b) X ,Y are positively correlated

c) r= 1 - 0.0028 d) none

22) In order to find the correlation coefficient between X and Y where n=12, ∑x=30, ∑y=5, ∑x²=670, y²=285, ∑xy=334. On subsequent verification it was found that the pair (11,4) was copied wrongly, the correct value being(10,14). find correct correlation coefficient.

a) -0.87 b) -0.78 c) 0.87 d) 0.78

23) In order to find the correlation coefficient between X and Y where n=25, ∑x=125, ∑y=100, ∑x²=650, y²=460, xy= 508. On subsequent verification it was found that (6,14) and (8,6) was copied wrongly, the correct value being(8,12) and (6,8). find correct correlation coefficient.

a) 0.76 b) 0.67 c) -0.756 d) -0.678

24) For bivariate data, there exist many types of relationship between the variables?

a) one b) two c) three d) more

25) If the data are collected for two variables simultaneously, it is known as:

a) Univariate Data.

b) Biveriate Data

c) Conditional Data

d) Marginal Data

26) The frequency distribution related to two variables collected at the same point of time is known as :

a) Bivariate Frequency Distri.

B) Joint frequency distribution

C) two-way frequency distribution

D) all of the above

27) From the Bivariate Frequency Distribution, we can obtain which of the following Univariate distribution ?

A) Marginal distribution

B) Conditional distribution

C) both A and B above

D) neither A nor B above

28) The total number of cells in (m+n) bivariate frequency table are :

A) m+n. B) m -n. C) m. D) mn

29) In a bivariate frequency distribution, if there m classification for x and n classification for y, then there would be altogether ___ conditional distribution.

A) m+n. B) m -n. C) m. D) mn

30) In (m+n) bivariate frequency table, the maximum number of marginal distribution is:

A) 1. B) 2. C) No marginal distribution exists. D) m+n.

31) In (m+n) bivariate distribution table, some of the cell frequencies may be:

A) Zero. B) negative.

C) Both A and B above. D) none

32) For which of the following statements the correlation will be negative ?

A) production and price unit.

B) sale of woolen garments and day temperature.

C) Only A above

D) Both A and B above

33) Which of the following is the example of variables Which are uncorrelated ?

A) Profit & Investment

B) Price & Demand of an item.

C) Shoe-size & Intelligence.

D) Yield & Rainfall.

34) Karl Pearson's correlation coefficient may be defined as:

A) The ratio of covariance between the two variables to the product of the standard deviations of the two variables.

B) The ratio of covariance between the two variables to the product of the variance of the two variables.

C) The ratio of product of standard deviations of the two variables to the covariance between the two variables.

D) none

35) If the value of "r" is "+". then there is a____correlation between the variables.

A) negative. B) positive

C) both negative and positive

D) neither positive nor negative.

36) If r=0, then there is____ correlation between the two variables.

A) negative. B) positive. C) No

D) both positive and negative.

37) If the variables are inversely proportional to each other, then statistically they are__ correlated.

A) positively. B) not

C) negatively. D) none

38) Which of the following statements are valid?

A) positive value of correlation between x and y implies that if x decreases, y tends increase.

B) Correlation coefficient is independent of the origin of reference but is dependent on the units of measurement.

C) Correlation coefficient between u and v turned out to be -1.02

D) none

39) Product moment Correlation coefficient, can measure Correlation when the variables are having which type (s) of relation?

A) non-linear. B) linear

C) curvilinear. D) both A and B

40) ___ is the simple way of the diagrammatic representation of bivariate data.

A) Time Series Graph.

B) Radar Diagram

C) Frequency Polygon

D) Scatter Diagram

41) The scatter diagram gives us the ___ ideas about the Correlation between the two variables

A) exact. B) Vague. C) No.D) none

42) Scatter Diagram can be applied for___ Correlation.

A) linear. B) Curvilinear

C) A as well as B above

D) Only B above

43) Scatter Diagram fails to:

A) Distinguish between different types of Correlation.

B) Measure the extent of relationship between the variables

C) Both A and B above. D) none

44) On a scatter diagram if one is not able to recognise a specific pattern, then the two variables depicts_____ Correlation.

A) no. B) positive. C) both negative and positive

D) perfectly negative.

45) Which of the following regarding value of "r" is TRUE ?

A) "r" is a pure number

B) "r" lies between-1 and+1

C) Only B is true

D) Both A and B are true

46) Which of the following regarding value of "r" is TRUE ?

A) 0 and 1. B) - 1 and 0

C) - 1 and+1. D) none

47) ____ Correlation coefficient is a measure of reliability.

A) standard error.

B) statistical error

C) probable error

D) both A and C above

48) Coefficient of Determination is:

A) square of Correlation

B) cube of Correlation

C) square root of Correlation

D) none

49) Standard Error of "r" is given by:

A) (1 - r²)/n. B) (1-r²)/√n

C) (1+r²)/√n. D) none

50) Probable error of r is how many times the standard error of r

A) 6.745.B) 67.C)0.6745 d) 0.4765

51) Coefficient of non-determination is :

A) r². B) 1+r². C) 1 - r². D) 1

52) The ratio of coefficient of non- determination and the square root of the number of variables is known as:

A ) Probable error. B) error

C) standard error. D) none

53) Find Correlation coefficient

X: 2 3 5 8 9

Y: 4 6 10 16 18

A) 1 B) 0 C) -1 D) none

54) If r ˣʸ = 0.6, what will be the value of rᵤᵥ , where u = 3x +5 and v= 4y -3

a) -0.6 b) 0.6 c) 0.36 d) none

55) If r ˣʸ = 0.6, what will be the value rᵤᵥ , where u= 3x+5 and v=-4y+3

a) -0.6 b) 0.6 c) 0.36 d) 0.66

56) U= 2X +11 and V= 3Y+7, What will be the Correlation coefficient between U and V ?

A) 1. B) -1. C) r ˣʸ. D) none

57) In a Correlation analysis, the values of Karl Pearson's Correlation coefficient and it's probable error were found to be 0.90 and 0.04 respectively. Find the value of n.

A) 9. B) 11. C) 10. D) 15

58) Which of the following methods to Calculate Correlation coefficient do not take into account the magnitude of the two variables?

A) Product moment method

B) Rank correlation Method

C) Concurrent deviation method

D) All of the above.

59) Which is the quickest method to find the Correlation coefficient between two variables?

A) Scatter Diagram

B) Method of concurrent deviation

C) Product Moment method

D) Spearman's Rank correlation

1) D 2) C. 3) D 4) c 5) b 6) A. 7) c. 8) d 9) c 10) b 11) b. 12) a 13) b 14) b 15) a. 16) b 17) c 18) a 19) a 20) a 21) d 22) d 23) b 24) c 25) b 26) d 27) c 28) d 29) a 30) b 31) a 32) d 33) c 34) a 35) b 36) c 37) c 38) d 39) b 40) d 41) b 42) c 43) b 44) a 45) d 46) d 47) c 48) a 49) b 50) c 51) c 52) c 53) a 54) b 55) a 56) c 57) c 58) c 59) a

Sunday, 24 May 2020

MATRIX (A- Z) For XII

A) DEFINITION

A matrix is defined as a rectangular arrangement of numbers in rows and columns. e.g.,

A = 2 √3 4

1 √2 0

B = 2 3 1 ≺ row 1

5 6 0 ≺ row 2

0 9 4 ≺ row 3

| | |

col. 1 col.2 col.3

NOTE:

1) A matrix is always denoted by a capital letters, like A, B, C, ...etc.

2) The number which are listed within brackets are known as

" Elements or Entries or Members" of the matrix.

3) Generally [ ] or ( ) brackets are used to donate a matrix.

4) The horizontal lines are known as " ROWS " and vertical lines are known as " COLUMNS ". Thus the matrix A has two rows and three columns. We say to be a rectangular (since, the number of rows is different from number of columns) matrix of order 2x3 and B to be a square matrix or order 3x3 (since, the number of rows is same as the number of columns).

• In other words we can say A matrix having m rows and n columns is called a matrix of order m x n (read as m by n).

a₁₁ a₁₂ ........ a₁ⱼ ....... a₁ₙ

a₂₁ a₂₂ ....... a₂ⱼ ....... a₂ₙ

.... .... ....... .... ....... ....

aᵢ₁ aᵢ₂ ....... aᵢⱼ ...... aᵢₙ

... ... ...... ... ...... ...

aᵤ₁ aᵤ₂ ...... aᵤⱼ ...... aᵤₙ

a₁₁ a₁₂ ........ a₁ⱼ ....... a₁ₙ

a₂₁ a₂₂ ....... a₂ⱼ ....... a₂ₙ

.... .... ....... .... ....... ....

aᵢ₁ aᵢ₂ ....... aᵢⱼ ...... aᵢₙ

... ... ...... ... ...... ...

aᵤ₁ aᵤ₂ ...... aᵤⱼ ...... aᵤₙ

The elements aᵢⱼ occurs in the ith row and jth column is called

(i j)-th element. For example, a₃₂ is the element of 3rd row and 2nd column.

The matrix written above is also denoted by the symbol (aᵢⱼ)ᵤₓₙ or [aᵢⱼ]ᵤₓₙ where i= 1, 2, 3, ....., m and j= 1, 2, 3, ....,n.

Different matrices are symbolised different capital letters as A, B, C etc.

REAL METRIX : if the elements of a matrix be all real then the matrix is called a real matrix.

B) SOME DEFINITION OF METRIX:

a) ROW MATRIX: A matrix whose elements are arranged in one row only is called a row matrix.

For examples, [a₁ a₂ a₃ ........aₙ] is a row matrix. Clearly, the order of the matrix is 1 x n.

b) COLUMN MATRIX: A matrix whose elements are arranged in one Column only is called a Column matrix. For example,

b₁

b₂

b₃

bᵤ is a column matrix. Clearly, the order of the matrix is m x 1.

c) NULL or ZERO MATRIX : A matrix whose every element is zero is called a null matrix or zero matrix. A null matrix is generally denoted by O. A null matrix of order u x n is denoted by Oᵤ ₓₙ

Example ) ( 0 0 0)

or 0 0 0 0 0

0 0 or 0 0 0

0 0 0

are all null matrix

d) SQUARE MATRIX: A matrix whose number of rows and columns are equal is called a square matrix. If number of rows = number of columns= n, then the matrix is called a square matrix of order n (or n x n square matrix). For example, a₁ b₁ c₁

a₂ b₂ c₂

a₃ b₃ c₃

is a square matrix of order 3 (or third order matrix).

DIAGONAL ELEMENTS: If A =[aᵢⱼ] be a square matrix, then elements where I = j are called diagonal elements and the straight line on which they lie is called the Principal diagonal or simply diagonal of the square matrix. For example, for the square matrix

1 3 5

4 6 8

2 7 4

is the diagonal elements are 1, 6, 4.

d) DIAGONAL MATRIX: A square matrix whose all the elements Except the elements in principal diagonal are zero is called a diagonal matrix. For example

a₁ 0 0

0 b₂ 0

0 0 c₃

is a diagonal matrix of order 3. And principal diagonal is a₁ b₂ c₃

2 0

0 3

is a diagonal matrix of order 2. And principal diagonal is 2 3.

NOTE:

1) A square matrix has two diagonals. The diagonal extending from left-hand top corner to right hand bottom corner is known as PRINCIPAL or MAIN or LEADING diagonal.

2) Atleast one element of the principal diagonal must be non-zero.

e) SCALAR MATRIX: A diagonal matrix whose elements on the principal diagonal are non-zero and all equal to each others is called scalar matrix for example

1) -2 0 2) 7 0 0

0 -2 0 7 0

0 0 7

f) UNIT or IDENTITY MATRIX: A scalar matrix whose elements on the principal diagonal are all 1 is called a unit or identify matrix. An unit matrix is generally denoted by I. Sometimes the order of the unit matrix is also written as suffix of I. For example. I₂ and I₃ are unit metrices of order 2 and 3 where

I₂ = 1 0 and I₃= 1 0 0

0 1 0 1 0

0 0 1

NOTE:

1) All the diagonal elements of the unit matrix should be + 1 only.

2) The diagonal matrix, scalar matrix and unit matrix all the square matrices i.e., in each of them the number of rows= the number of columns.

g) UPPER TRIANGULAR MATRIX:

This is a square matrix whose each element below the princpal diagonal is zero. Example

1) 2 3 2) 2 4 0 3) 2 0 0

0 1 0 3 1 0 0 1

0 0 5 0 0 5

h) LOWER TRIANGULAR MATRIX:

This is a square matrix whose elements above the principle diagonal is zero. Example.

1) 2 0 2) 2 0 0 3) 0 0 0

1 3 3 1 0 2 0 0

4 0 1 3 1 0

****************"*"*********""""***********

* ADDITION OF TWO MATRICES *

Two matrices can be added or subtracted only if they are of the same order and this is done by adding or subtracting the corresponding elements of two matrices.

PROPERTIES OF MATRIX ADDITION:

1) A+ B= B + A

2) A +(B+C)= (A+B)+C

3) A + O= O +A = A

Where A, B, C are the matrices of the same order and O is the null matrix of the same order.

Exercise - A

1) If A= 2 5 and B= 1 -3

-3 7 2 5

Find A+ B. 3 2

-1 12

2) If A= 3 -6 & B= 1 -3

9 3 3 0

find 2A+ B. 7 -15

21 6

3) If A= 2 -3 and B= -1 6

4 5 3 2

Find 2A+ 5B. -1 24

23 20

4) If M= 2 0 and N= 2 0

1 2 -1 2

Find M+ 2N. 6 0

-1 6

5) If A= 2 0 and B= 1 2

-3 1 3 1

Find 3A+ 4B. 10 8

3 7

6) If A= 1 4 and B= -4 -1

2 3 -3 -2

Find 2A+ B. -2 7

1 4

7) If A= 2 4 and B= 1 3

3 2 -2 5

Find 2A+ 3B. 7 17

0 19

8) If A = -3 5 and B= 2 -3

-9 11 6 7

find 2A + 3B. 0 1

0 1

9) If A= -3 2 1 & B= 3 5 1

1 -4 7 -1 4 -2 find A+ B. 0 7 2

0 0 5

10) If A= diag(1 -1 2) and B= diag(2 3 -1), find

A) A+ B. diag (3 2 1)

B) 3A + 4B. diag(11 9 2)

11) If M= 2 0 and N= 2 0

1 2 -1 2 find

M +N. 4 0

0 4

Exercise - B

1) If A= 3 -1 7 & B= 2 -2 -1

0 1 2 1 2 3

Find A+ B. 5 -3 6

1 3 5

2) If A= 0 2 3 & B= 7 6 3

2 1 4 1 4 5

Find 2A+ 3B. 21 22 15

7 14 23

3) 3 -1 0 0 = 3 -1

If A= 2 1 & B= 0 0 2 1

4 -2 0 0 4 -2

Find A+ B. 3 -1

2 1

4 -2

4) if A = 1 2 3 B= 4 -2 -2

0 3 - 5 6 2 -1

1 0 0 1 -2 3

find 2A + 3B. 14 -2 0

18 12 -13

5 -6 9

5) 1 2 3 3 -1 3

A= 2 3 1 & B = -1 0 2

Find 2A + 3B. 5 1 15

1 6 5

6) A= 2 3 4 & B= 3 0 5

0 4 6 5 3 2

5 8 9 0 4 7 find 3A - 2B. 0 9 2

-10 6 14

15 16 13

7) A=1 -3 2 B= 2 -1 -1 C= -3 4 -1

2 0 2 1 0 -1 -3 0 -1

Find 2A + 3B+ C

8) If A=1 3 5 and B= 4 2 3

0 4 1 1 3 -7

Verify 2(A+B)= 2A +2B

* SUBSTRACTION OF MATRICES *

If A and B are metrices of the same order, then A - B = A+ (-1)B. Hence if A= [aᵢⱼ]ᵤₓᵥ and B= [bᵢⱼ]ᵤₓᵥ then

A - B = [aᵢⱼ]ᵤₓᵥ + (-1) [bᵢⱼ]ᵤₓᵥ

= [aᵢⱼ - bᵢⱼ]ᵤₓᵥ

Thus A - B is a matrix of the same order of matrix A and B, whose elements are the members obtained from substracting the corresponding elements of B from A. for example, if

A= x₁ y₁ z₁ a₁ b₁ c₁

x₂ y₂ z₂ and B= a₂ b₂ c₂

x₃ y₃ z₃ a₃ b₃ c₃

then,

A - B= x₁- a₁ y₁-b₁ z₁ - c₁

x₂ -a₂ y₂- b₂ z₂ -c₂

x₃ -a₃ y₃ -b₃ z₃ -c₃

NOTE:

1) Since - A= (-1)A = (-1) [aᵢⱼ]ᵤₓᵥ i.e., all the elements of A with sign changed, hence - A is called the negative matrix of matrix A.

Again, since A +(- A)= (- A)+ A=0, so - A is called the additive inverse of A.

2) If A + B = A + C, or

B + A = C + A, then B = C, where A, B, C are matrices of same order.

Exercise - C

1) If A = 2 4 and B = 3 6

5 6 5 4

Find A- B. -1 -2

0 2

2) If A= 3 -6 & B= 1 -3

9 3 3 0

find A-2 B. 1 0

3 4

3) If M= 2 0 and N= 2 0

1 2 -1 2

Find:

A) 2M - 3N. -2 0

5 -2

B) 2(M - 3N). 16 0

8 -8

C) 2M - N. 2 0

3 2

D) 7M - 5N. 4 0

12 4

4) If A= 7 6 3 and B = 0 2 3

1 4 5 2 1 4

Find 3B - A. 21 16 6

1 11 11

5) if A= 1 2 & B= -2 3

-2 3 1 2

find 7A - 5B. 17 -1

-19 11

6) If A= diag(2 -5 9), B= diag(1 1 -4) and C=(-6 3 4) find

i) A- 2B. diag(0 -7 17)

ii) B+C - 2A. diag(-9 14 -18)

iii) 2A + 3B-5C. diag(37 -22 -14)

Exercise - D

1) If A = 2 4 and B = 3 6

5 6 5 4

find X if 3A + 4B - 2X= 0. 9 18

35/2 22

2) If A= 5 -4 & B= 3 -2

-11 6 -1 4 find the Matrix X when A+ 2X = 3B. 2 -1

4 3

3) If B= -2 2 0 C= 2 0 -2

3 1 4 7 1 6 find the value of A as the relation 2A- 3B + 5C = 0. -8 3 5

-13 -1 -9

4) If A= diag(2 -5 9), B= diag (1 1 -4) and C= diag (-6 3 4) find

A) A- 2B. diag(0 -7 17)

B) B+ C - 2A. diag(-9 14 -18)

C) 2A + 3B - 5C. diag(37 -22 -14)

5) if A= 1 2 B= -2 3 C= 0 3

-2 3 1 2 2 -1

find

i) A + 2B - 3C. -3 -1

-6 10

ii) 3A - 4B + C. 11 0

-6 -1

iii) A - 3B + 2C. 7 -1

-1 -5

6) Let A= -1 0 2 B=0 -2 5 & C= 1 -5 2

3 1 4 1 -3 1 6 0 -4 Compute 2A - 3B + 4C. 2 -14 -3

27 11 -11

7) if A = 1 2 3 & B= 4 -2 -2

0 3 - 5 6 2 -1

1 0 0 1 -2 3

a) If 3A + X = B find X. 1 -8 -11

6 -7 14

0 -2 3

b) if X - 3B = 2A find X. 14 -2 0

18 12 -13

5 -6 9

8) If A - 2B= -7 7 & A - 3B= -11 9

4 -8 4 -13 find the Matrix A and B.

** EQUALITY OF TWO MATRICES:

Two matrices are said to be equal if

1) they are of same orders.

2) each element of one is equal to the corresponding element of the other. For example, if A and B are matrices both of order 2x 3 and

A = a b c and B = p q r

d e f x y z

Then A= B implies a= p, b= q, c= r, d= x, e= y, and f= z.

EXERCISE-E

Type-2

1) x 7 + 6 -7 = 20 7

9 y-5 4 5 22 15

find x and y. 14, 22

2) a 3 + 2 1 - 1 b = 5 0

4 2 1 -2 -2 c 7 3

find a, b, c. 4, -1,3

3) If A= 2a - b 4 = 2 4

3 a+2b 3 6 find the value of a and b. 2,2

4) If x+y 2 = 6 2

5 xy 5 8 find the value of x and y. 2, 4 or 4, 2

5) If A=2 a B = -2 3 C = c 9

-3 5 7 b -1 -11

and 5A + 2B = C find a, b, c. 6/5, -18,6

6) If A= x y B= 1 -1 & C= 3 5

z t 0 2 4 6 with the relation 2A+ 3B = 5C then find the value of x,y,z and t. 6,14,10,12

7) x -y y+ z = t - x t - z

2+ t 3+ t z+ y 6 find x,y,z,t. 5,7,-2,3

8) y+ z x +z = 9 - t 8- t

7 - t 6 - z x+ y x+ y find the value of x,y z. 1,2,3,4

9) x - y 2z+ w = 5 3

2x- y 2x+ w 12 15 find the value of x, y, z, w. 7,2,1,1

10) 2x+1 3y = x+3 y²+2

0 y²-5y 0 -6 find the value of x and y. 2,2

11) x+3 z+4 2y-7 = 0 6 3y-2

4x+6 a-1 0 2x -3 2c-2

b-3 3b z+ 2c 2b+4 -21 0 find the value of a, b, c, x, y z. -2, -7, -1, -3, -5, 2

12) If A= 2 0 -4 B= 1 -2 0 C= 2

1 4 2 2 -1 3 0

1 Find the Matrix C with the relation (3A - 2B)C = D. -4

-2

13) A= 1 2 3 B= 1 0 2 C= 4 4 10

-1 -3 2 3 4 5 4 2 14 find the value of k if 2A+ kB=C. 2

Type- 2

14) If X + Y = 7 0 and X - Y= 4 -1

2 3 5 -2

find metrices X And Y.

11/2 -1/2 3/2 1/2

7/2 1/2 -3/2 5/2

15) A+ B= 2 2 2A+ 3B= 5 4

0 2 and 0 5 then find the Matrix A and B.

1 2 1 0

0 1 0 1

16) If A+B= 1 2 0 A - B= 3 0 2

3 5 4 1 1 0

Find the metrices A and B

2 1 1 -1 1 -1

2 3 2 1 2 2

17) If A+B= 2 2 and A - B= 5 4

0 2 0 5

Find the metrices A and B.

7/2 3 -3/2 -1

0 7 0 -3/2

18) If 2P + Q= 4 5 & P + 2Q= 2 4

3 8 3 1 Then find the value of P+ Q. 2 3

2 3

19) If 2A+ B= 4 4 7 & A- 2B = -3 2 1

7 3 4 1 -1 2 find the Matrix A and B

1 2 3 2 0 1

3 1 2 1 1 0

20) If 2x -y= 6 -6 0 & x +2y =3 2 5

-4 2 1 -2 1 -7 find the Matrix x and y. 3 -2 1 0 2 2

-2 1 -2 0 0 -3

21) If 2A+B= 4 7 16

7 -3 12

. 13 6 2

And 3B - A= 5 0 13

7 -2 8

11 4 -1

Find metrices A and B

2 1 6 1 3 5

3 -1 4 2 -1 4

5 2 0 4 2 1

22) If A + 2B = 1 2 0

6 -3 3

-5 3 1

and 2A - B= 2 -1 5

2 -1 6

0 1 2

Find metrices A and B.

0 5/3 -5/3 1 1/3 10/4

10/3 -5/3 0 -2/3 1/3 3

-10/3 5/3 0 5/4 -1/3 1

23) X - y = 1 1 0 and x+ y= 3 5 1

1 1 0 -1 1 4

1 0 0 11 8 9 find Matrix x and y. 2 3 1 1 2 0

0 1 2 -1 0 2

6 4 0 5 4 0

MULTIPLICATION OF A MATRIX BY A SCALAR :

Multiplying a matrix by a real number is called multiplication of a matrix by a scalar.

If k is a real number (scalar) and A be a matrix, then their product kA is defined by the matrix of the same order as A, whose elements at each position is k times that of A. i.e., if A = [aᵢⱼ)ᵤₓᵥ, thenkA =ln([kij]ᵤₓᵥ

PROPERTIES OF MULTIPLICATION OF A MATRIX BY A SCALAR:

Let p,q are any two scalar and A and B are metrices of the same order then

PROP 1). (p+q)A= pA + qA

PROP 2) p(A+B)= pA + pB.

PROP 3) p(qA) = (p q)A or

(p q)A = p(qA)

Exercise - F

1) If A = 1 3 B = 4 -5

2 -2 3 -1 Find

i) AB. 13 -8

2 -12

ii) BA. -6 22

1 11

iii) A². 7 -3

-2 10

iv) (AB)². 153 8

2 128

v) A² - B². 6 12

-11 24

vi) A² - 2B. -1 7

-8 12

2) If A = 1 2 B= 3 4 C= -1 0

5 -4 0 2 2 -2

find i) ABC

ii) (A+B)C

iii) A² - BC

iv) AC + B²

v) (A+ B) (A - B)

vi) AC + B²

3) If A= 3 5 & B= 1 6

1 -2 -4 3 show AB≠ BA

4) A= 2 0 & B= 3 0

3 1 3 2 show AB = BA

5) A= 2 1 & B= 1 -2

1 2 -1 2 show AB = B

6) If A = 1 2 3 and B = 6 -2 -3

1 3 3 -1 1 0

-1 0 1 1 2 4

evaluate AB , BA

7) If A = 1 -2 1

-1 2 -1 show A² = A

-2 4 -2

8) If A= 1 -1 and B= 1 1

-1 1 1 1

prove AB=0.

9) If A = 1 2 1 & B= 1 4 0

1 -1 1 -1 2 2

2 3 1 0 0 2

Find AB - 2B

10) If A= 3 2. 5 B= 1 2 & C= 7 - 8

2 -4 0 2 -1 5 Find AB - C

11) If A= 0 1 2 B = 2 1 3

1 2 3 -1 0 1

3 1 1 3 -1 4

Verify AB≠ BA

12) If A = 1 2 3 and B= 6 -2 -3

1 3 3 -1 1 0

1 2 4 -1 0 1

Check AB = BA

13) If A=2 -3 -5 and B= -1 3 5

-2 4 5 1 -3 -5

1 -3 -4 -1 3 5

Prove AB≠BA

14) If A= 1 & B= 1 2 3 find BA

15) If A is 3 -2 0 & B= 2 find BA

16) If A= 2 3 -1 B= 1 & C= 1 -2

3 0 2 1

Verify A(BC)≠ (AB)C

17) if A= 0 1 0 & B = 0 0 1

0 0 1 1 0 0

1 0 0 0 1 0

verify i) A²= B ii) B² = A

18) If A= 2 -1

- 1 2 show that A²- 4A+ 3I = 0, where I is 2 x2 unit Matrix and 0 is 2x2 zero Matrix.

19) If I= 1 0 & B= 3 2

0 1 2 1 show that A²- 4A - I= 0, where 0 is the zero Matrix of order 2

20) If A= 4 3 and B= 1 0

2 5 0 1 find the values of x and y so that A² - xA + yI =0, where 0 is the zero Matrix of order 2. 9, 14

21) If A = 2 -1 show A² - 4A + 3I =0

-1 2

22) If A = 2 0 1 find A² - 5A + 6I

2 1 3

1 -1 0

23) If A = a b

c d

show A² - (a+d)A+(ad-bc) I=0

24) for what values if x,y,z If A .A = I′ &

A= 0 2y z

x y -z

x -y z

25) A= 1 -1 and B = 1 x

2 -1 4 y and

(A +B)² = A² + B² then find x,y.

26) if A= 6 5 & C= 11 0

5 6 0 11 find Matrix B such that AB= C. 6 -5

-5 6

MATRIX NOTATION:

A₂ ₓ ₃ = a₁₁ a₁₂ a₁₃

a₂₁ a₂₂ a₂₃

The above notation means that A is a matrix having two rows and three columns. a₁₁ is the element situated at the junction of 1st row and 2nd Column, a₂₃ is the element situated at the junction of 2nd row and 3rd Column, etc.

Example Let B₃ₓ₂ = 4 -1

0 2

1 4

then b₁₁ = 4, b₁₂ = -1, b₂₁=0, b₂₂= 2,

b₃₁ = 1, b₃₂= 4.

NOTE : A matrix A having m rows and n columns is also denoted (aᵢⱼ)ᵤₓₙ .

Example:

Construct a 2x 2 matrix A = (aᵢⱼ) whose elements are given by aᵢⱼ=(1+2j)²/2

Solution)

A= a₁₁ a₁₂

a₂₁ a₂₂

Where, a₁₁= (1+2x1)²/2= 9/2

a₁₂ = (1+2x2)²/2 = 25/2

a₂₁ = (2+ 2x1)²/2= 8

a₂₂ = (2+ 2x2)²/2= 18

So required matrix A= 9/2 25/2

8 18

Exercise - G

1) construct a 2 x 2 matrix A = [aᵢⱼ] whose elements are given by

A) aᵢⱼ = (i+j)²/2. 2 9/2

9/2 8

B) aᵢⱼ = (i -j)²/2. 0 1/2

1/2 0

C) aᵢⱼ= (i- 2j)²/2. 1/2 9/2

0 2

D) aᵢⱼ= (2i-j)/2. 1/2 0

3/2 1

E) aᵢⱼ= |2i -3j)|/2. 1/2 2

1/2 1

F) aᵢⱼ= |-3i+ j|/2. 1 1/2

5/2 2

2) construct a 2x2 Matrix A = [aᵢⱼ] whose elements are given by aᵢⱼ= (i - j)/(i + j).

0 -1/3

1/3 0

3) Construct a 2x3 Matrix whose elements aᵢⱼ are given by:

A) aᵢⱼ= i. j. 1 2 3

2 4 6

B) aᵢⱼ= 2i - j. 1 0 -2

3 2 1

C) aᵢⱼ= i+ j. 2 3 4

3 4 5

4) Construct a 2x2 Matrix whose elements aᵢⱼ are given by:

A) (i+j)²/2. 2 9/2

9/2 8

B) (i -j)²/2. 0 1/2

1/2 0

C) (2i+j)²/2. 9/2 8

25/2 18

D) aᵢⱼ= |-3i+ j|/2. 1 1/2

5/2 2

5) Construct a 2x3 matrix A= [aᵢⱼ] whose elements are given by

A) aᵢⱼ= (i - j)/(i+j). 0 -1/3 -1/2

1/3 0 -1/5

B) aᵢⱼ= i. j. 1 2 3

2 4 6

C) aᵢⱼ = 2i - j. 1 0 -1

3 2 1

D) aᵢⱼ = i +j. 2 3 4

3 4 5

E) aᵢⱼ = (i+j)²/2. 2 9/2 8

9/2 8 25/2

6) Construct a 3x4 matrix A= [aᵢⱼ] whose elements aᵢⱼ are given by:

A) aᵢⱼ= i+ j. 2 3 4 5

3 4 5 6

4 5 6 7

B) aᵢⱼ= i - j . 0 -1 -2 -3

1 0 -1 -2

2 1 0 -1

C) aᵢⱼ= 2i. 2 2 2 2

4 4 4 4

6 6 6 6

D) aᵢⱼ= j. 1 2 3 4

1 2 3 4

E) aᵢⱼ = 1/2 |-3i+j|. 1 1/2 0 1/2

5/2 2 3/2 1

4 7/2 3 5/2

7) Construct a 4x3 matrix elements

A) aᵢⱼ= 2i + i/j. 3. 5/2. 7/3

6 5 14/3

9 15/2 7

12 10 28/3

B) aᵢⱼ=(i-j)/(i+j). 0 -1/3 -1/2

1/3 0 -1/5

1/2 1/5 0

3/5 1/3. 1/7

Exercise - H

1)If A = 2. 4 -1 and B = 3 4

-1 0 2 -1 2

2 1

Find

i) A′

ii) A′ +B′

iii) (A - B)ᵗ

iv) (AB)′

v) Aᵗ Bᵗ

2) If A= 2 1 and B= 1 -2

3 4 -1 1

Prove (AB)'= B ' A '

3) If A = 0 -1

2 3

Prove (A ')' = A

4) If A = 4 2 -1 and B= 2 3

3 -7 2 -3 0

1 5

Find possible or not A+B, A - B , AB, BA

5) Let A= 2 -3 & B= 1 0

-7 5 2 -4 Verify

A) (2A)'= 2A'

B) (A+ B)'= A' + B'

C) (A - B)'= A' - B'D) (AB)' = B' A'

6) Let A=1 -1 0 & B= 1 2 3

2 1 3 2 1 3

1 2 1 0 1 1 verify

A) (A+ B)'= A'+ B'

B) (AB)'= B' A'

C) (2A)'= 2A'

7) If A= 2 1 3 & B= 1 -1

4 1 0 0 2

5 0 verify

(AB)'= B' A'

Exercise - I

** A square Matrix A is a symmetric Matrix iff A'= A

** A square Matrix A is a skew-symmetric Matrix iff A'= - A

*** Sum of a symmetric and skew-symmetric Matrix= 1/2 (A+ A')+ 1/2 (A - A')

1) If A= 3 -1 1

-1 2 5

1 5 -2 is a symmetric Matrix.

2) if A= 0 2 -3

-2 0 5

3 -5 0 is a skew-symmetric Matrix.

3) If A= 2 3

4 5 Prove A- A' is a skew-symmetric Matrix

4) if A= 3 -4

1 -1 show that A - A' is a skew-symmetric Matrix.

5) If A= 5 2 x

y z -3

4 t -7 is a symmetric Matrix, find x,y z, t. 4, 2 , -3

6) Express the Matrix A as the sum of symmetric and skew-symmetric Matrix

A= 4 2 -1

3 5 7

1 -2 1

7) Express the Matrix A as the sum of symmetric and skew-symmetric Matrix

A= 3 -4

1 -1

8) Let A= 3 2 7

1 4 3

2 5 7 Find matrix X and Y such that X+Y = A, where X is a symmetric and Y is a skew-symmetric matrix.

9) If A= 2 4

5 6 Prove A+A' is a symmetric matrix where A' is the transpose of A.

EXERCISE -J

Find the determinants of

1) 2 -3

4 1 14

2) 4 2 -3

1 4 4 4

3) 2 3 -1

4 1 2

3 1 -1 23

4) 1 0 1

2 -1 1

3 0 1

5) 2 3 -4

1 3 -1

3 1 4 37

***Evaluate Expanding by 2nd row:

6) 2 0 -1

1 3 2

-2 4 1 -20

2) 3 -2 1

1 0 1

2 4 5 2

EXERCISE - K

A) Find the cofactors of a₂₁ & a₂₂

1) A= 2 -3

1 4 3, 2

2) -4 1

3 -2 -1,-4

B) Find the cofactors of a₁₃, a₂₂, a₃₂

1) 0 -1 2

A= -3 4 -5

6 -7 8 -3,-12,-6

2) 2 4 -6

A= -8 4 3

2 -4 1 24, 14, 42

EXERCISE - L

Find the adjoint of the following:

1) a b d -b

c d -c a

2) -3 5 4 -5

2 4 -2 -3

3) p q s -q

r s -r p

4) - 2 3 4 -3

- 5 4 5 -2

5) 1 -3 1 3

2 1 -2 1

6) cosx sinx cosx - sinx

sinx cosx -sinx cosx

7) 1 tan(x/2) 1 -tan(x/2)

- tan(x/2) 1 tan(x/2) 1

8) 1 2 2 -3 2 2

2 1 2 2 -3 2

2 2 1 2 2 -3

9) 1 2 5 2 3 -13

2 3 1 -3 6 9

-1 1 1 5 -3 -1

10) 2 -1 3 -22 11 -11

4 2 5 4 -2 2

0 4 -1 16 -8 8

11) 2 0 -1 3 -1 1

5 1 0 -15 7 -5

1 1 3 4 -2 2

12) cosx - sinx 0 cosx sinx 0

sinx cosx 0 -sinx cosx 0

0 0 1 0 0 1

13) 1 1 1 9 -1 4

2 1 -3 -3 4 5

-1 2 3 5 -3 -1

14) 2 -1 3 -22 11 -11

4 2 5 4 -2 2

0 4 -1 16 -8 8

15) -6 0 3 7 -15 -3

-2 1 1 0 0 0

-4 -5 2 14 -30 -6

16) -4 -3 -3

If A= 1 0 1

4 4 3

Show that adj A= A

17) 1 -2 3

If A= 0 2 -1

-4 5 2

Find A(adj). 25 0 0

0 25 0

0 0 25

EXERCISE - M

State which of the following are INVERTIBLE :

1) 2 -3

2 3 YES

2) 4 -1

-4 1. NO

3) 5 2 -3

4 -5 2 NO

0 3 -2

4) 1 1 -1

2 3 1. YES.

1 -1 -2

*** FIND the condition for which the following matrices are INVERTIBLE

1) a b

c d ad-bc≠ 0

2) sinx cosx

-sinx cosx (0≤0≤π/2). 0≠0,π/2

Find the inverse of following:

1) 2 -1 4/11 1/11

3 4 -3/11 2/11

3) cosx sinx cosx -sinx

-sinx cosx sinx cosx

4) 0 1 0 1

1 0 1 0

5) a b (1+bc)/a -b

c. (1+bc)/a - c a

6) 2 5 1/17 -5/17

-3 1 3/17 2/17

7) 1 tanx cos²x sinx cos

-tanx 1 sinx cosx cos²x

8) If A=3 2 & B= 6 7

7 5 8 9 then find the value of (AB)⁻¹. -47 39/2

41 -17

9) 1 3 3 7 -3 -3

1 4 3 -1 1 0

1 3 4 -1 0 1

10) 1 2 3 -5/18 1/18 7/18

2 3 1 1/18 7/18 -5/18

3 1 2 7/18 -5/18 1/18

11) 1 2 5 4/27 17/27 3/27

1 -1 -1 -1/27 11/27 2/9

2 3 -1 5/27 1/27 -1/9

12) 2 0 -1 3 -1 1

5 1 0 -15 6 -5

0 1 3 5 -2 2

13) 1 0 0 1 0 0

0 cosx sinx sinx cosx 0

0 sinx -cosx 0 sinx -cosx

14) 0 0 -1 -2 1 1

3 4 5 14 -1/2 -3/4

-2 -4 -7 -1 0 0

*** FIND A If:

1) 2 - 1 2 1

A⁻¹= -3 2 3 2

2) 2 -1 3 -2 2 4

A⁻¹=1 1 1 0 1 -1

1 -1 1 2 -1 -3

EXERCISE -N

Prove that

1) a) A=2 3

5 -2 then A⁻¹ = A/19

2) if A=. 4 5

2 1 Verify A= 5I +6A⁻¹

3) show A = 2 -1 and B = 2

4 3 -3

and AX = B Find the matrix X.

4) If A= 1 -2 & B= 6 0

1 4 0 6 Find the Matrix C If CA = B. 4 2

-1 1

5) If A= 2 1 B= -3 2 & C= 1 0

3 2 5 -3 0 1 find Matrix X if AXB= C. 1 1

1 0

6) If A= 1 -4 B= -16 -6

3 -2 7 2 find the Matrix X if AX= B. 6 2

11/2 2

7) A= 5 4 & B= 1 -2

1 1 1 3 find the Matrix X as the relation AX = B. -3 -14

4 17

8) If A= 3 2 B= -1 1 & C= 2 -1

7 5 -2 1 0 4 find the Matrix X as AXB= C. -16 3

24 -5

9) If A= 5 3 & B= 14 7

-1 -2 7 7 find the Matrix X as XA= B. 3 1

1 -2

10) if A= 2 -1

-1 2

satisfies the relation A² - 4A + 3I =0 hence find A⁻¹

11) If A = 1 1

2 3

prove A² - 4A +5I=0. Hence find A⁻¹.

3/5 1/5

-2/5 1/5

12) If A = 4 5 satisfies A² - I=10A

5 6 Hence find A⁻¹

13) A= 2 -3

3 4 satisfies the equation x²- 6x+17= 0. Hence find A⁻¹. 4/17 3/17

-3/17 2/17

14) If A= 3 2

2 1

Prove A² - 4A - I= 0 where I= 2x2 metrics and 0 is Null metrics. Then find inverse of A.

15) If A= 2 -1

-1 2

Prove A² - 4A + 3I = 0 where I= 2x2 metrics and 0 is Null metrics.

16) A= 2 3

1 2 verify A²- 4A + I= 0, then find A⁻¹. 2 -3

-1 2

17) A= 3 1

7 5 find x and y so that A² + xI = yA. Hence, find A⁻¹. 8,8 5/8 -1/8

-7/8 3/8

18) A= 3 2

1 1 find a and b so that A² + aA+ bI = 0. Hence, find A⁻¹. -4,1 1 -2

-1 3

19) If A= 4 5

2 1 find A⁻¹ and show that 2A⁻¹ =9I - A.

20) If A= 4 3

2 5 find x and y such that A² - xA + yI = O. Hence, find the value of A⁻¹.

9,14 5/14 -3/14

-1/7 2/7

21)A= 1 2 2 satisfies A²-4A-5I=0

2 1 2 and, Hence find A⁻¹

2 2 1 3/5 2/5 2/5

2/5 -3/5 2/5

2/5 2/5 -3/5

22) IfA= 2 2 0 then A³-13A+12I=0

2 1 1 . Hence find A⁻¹

-7 2 -3

23) A= 1 2 3

3 -2 1 show A³ -23A-40I=0

4 2 1 Hence find A⁻¹

24) If A= 1 0 -2

-2 -1 2

3 4 1 Show that A³- A²- 3A - I = O. Hence find A⁻¹. -9 -8 -2

8 7 2

-5 -4 -1

25) If A = 1 0 -2

2 2 4

0 0 2

Find A² - 3A + 2I where I= 3x3 then find Inverse of A.

EXERCISE -O

Find the inverse of each of the following matrices by using elementary row transformation:

1) 5 2 1 -2

2 1 -2 5

2) 1 6 5/23 -6/23

-3 5 3/23 1/23

3) 7 1 3/25 1/25

4 -3 4/25 -7/25

4) 3 10 7 -10

2 7 -2 3

1) 0 1 2 1/2 -1/2 1/2

1 2 3 -4 3 -1

3 1 1 5/3 -3/2 1/2

2) 2 0 1 3 -1 1

5 1 0 -15 6 -5

0 1 3 5 -2 2

3) 2 3 1 1 1 -1

2 4 1 -1 1 0

3 7 2 2 -5 2

4) 3 -3 4 1 -1 0

2 -3 4 -2 3 -4

0 -1 1 -2 3 -3

5) 2 -1 4 -2 1/2 1

4 0 2 11 -1/2 -2

3 -2 7 4 -1/2 -2

EXERCISE - P

Solve:

A) 5x - 7y =2, 7x - 5y =3,. 11/24, 1/24

B) x - 2y -4 =0, -3x +5y+7 =0, -6,-5

C) 3x+ 4y =5, x-y =-3. -1,2

D) ax+ by=c, a²x +b²y =c².

E) 3/x - 5/y =1 , 2/x +3/y = 7.

F) a/x -b/y =a , a/y - b/x =b.

G) 5x +7y =-2, 4x +6y+3=0. 9/2,-7/2

H) 5x +2y =3, 3x +2y=5. -1,4

I) 3x +7y =4, x +2y+1=0. -15,7

A) x+y-z=3, 2x+3y+z=5 , 3x-y-7z=1. 3,1,1

B) x+ 2y+z=7; x+ 3z=11; 2x-3y=1. 2,1,3

C) 2y-3z=0, x+3y= -4, 3x+4y =3

E) x+y-6z=0, -3x+y+2z=0, x-y+2z=0

F) x+y+z=4, 2x-y+2z=5 , x-2y-z=-3

F) (a+b)x - (a-b)y= 4ab,

(a-b)x + (a+b)y = 2(a² -b²)

G) 1/x +2/y+ 3/z=2

2/x +4/y +5/z =3

3/x +5/y+6/z= 4

H) 2/x +3/y +2=0

5/y - 2/z -4 =0

3/z +4/x +7=0

Continue.......

Mg. A- R.1

1) If A = 2 -1

-1 2 and I is the unit Matrix of order 2, then A² is:

a) 4A -3I b) 3A -4I c) A - I d) A + I

2) The multiplicative inverse of 2 1

7 4

a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1

-7 -2 -7 2 7 2 7 -2

3) A is a square Matrix such that A³ = I; then A⁻¹ is:

a) A² b) A c) A³ d) none

4) Assuming that the sums and products given below are defined, which of the following is not true for Matrices?

a) AB = AC does not imply B= C

b) A + B = B+ A

c) (AB)' = B'A'

d) AB = 0 implies A=0 or B= 0

5) If A= 1 0 2 B= 5 a -2

-1 1 -2 1 1 0

0 2 1 -2 -2 b then the value of a and b are:

a) -4,1 b) -4 , -1 c) 4 ,1 d) 4 , -1

6) If A= -1 0

0 2 then the value of A³ - A² is

a) I b) A c) 2A d) 2I

7) If A= -x -y

z t then transpose of adj A is

a) t z b) t y c) t - z d) none

-y -x - z -x y -x

8) If A is a square Matrix of order 3x3 and $ is a scalar, then adj($A) is equal

a)$ adj A b) $²adj A c) $³adj A d) $⁴adj A

9) If A=3 5 B= 1 17

2 0 0 -1 then |AB| =?

a) 80 b) 100 c) -110 d) 92

10) The inverse of the Matrix 5 -2

3 1

a)-2/13 5/13 b)1 2 c) 1/11 2/11 d)1 3

1/13 3/13 -3 5 -4/11 5/11 -2 5

Mg. A- R.2

1) If A is a singular Matrix of order n then A. (adj A) is equal to

a) a null Matrix b) a row Matrix c) a column Matrix d) none

2) If A and B are two matrices and A⁻¹ and B⁻¹ exist, then (AB)⁻¹ is equal to

a) A⁻¹B⁻¹ b) AB⁻¹ c) A⁻¹B d) B⁻¹ A⁻¹

3) If A= 3 -5

-4 2 then the value of A² - 5A is

a) I b) 14I c) O d) none

4) If A= 5 6 -3

-4 3 2

-4 -7 3 then the co-factors of the elements of second row are:

a) 3,3,11 b) 3,-3,11 c) -39,3,-11 d) 39,-3, 11

5) If A= 1 2 & B= 1 2

2 3 2 1

3 4

Then

a) both AB and BA exist

b) neither AB nor BA exist

c) AB exists but BA does not exist

d) AB does not exist but BA exists

6) If A= 2 -1 B= 1 0

0 1 -1 -1 then (A+ B)² is not equal to

a) A² + AB+ BA+ B²

b) A² + AB+ BA+ B²I

c) A²I + AB+ BA+ B²

d) A² + 2AB+ B²

7) If A be an nx n Matrix and k any scalar, then det kA is equal to

a) k detA b) nᵏdetA c) kⁿdetA d) kn detA

8) If A= 1 2

3 -5 then A⁻¹ is:

a)-5 -2 b) -5/11 -2/11 c) 5/11 2/11 d)5 2

-3 1 -3/11 1/11 3/11-1/11 3 -1

9) If A= -1 2 & B= 5

2 -1 7 and AX= B, then X is

a) 19 17 b) 19/3 c) 19/3 17/3 d) 19

17/3 17

10) If A≠ O and B≠O are two matrices such that AB= O, then which of the following is correct?

a) detA= 0 or det B= 0

b) detA= 0 and det B= 0

c) detA=det B ≠ 0 d) none

Mg. A- R.3

1) If A is a square Matrix of order 3x3 and k is a scalar, then adj(kA) is equal to which of the following?

a)k adj A b)k² adj A c) k³ adj A d)2k adj A

2) If A= 0 1 2

1 2 3

3 1 1 and it's inverse B=[bᵢⱼ], then the element b₂₃ of Matrix B is

a) -1 b) 1 c) -2 d) 2

3) If A= a₁₁ a₁₂ a₁₃ B= 1 2 3

a₂₁ a₂₂ a₂₃ 2 3 4

a₃₁ a₃₂ a₃₃ 3 4 5

C= -1 -2 D= -4 -5 -6

-2 0 0 0 1

0 -4

with the relation A= BCD, then the value of a₂₂ is

a) 40 b) -40 c) -20 d) 20

4) If A= 1 2 & B= 3 8

3 4 7 2 with the relation 2X+ A = B, then the Matrix X is:

a) 2 -6 b) 1 -3 c) 1 3 d) 2 -6

4 -2 2 -1 2 -1 4 -2

5) If A= a 2

2 a and |A³|= 125, then a is

a) ±2 b) ±3 c) ±5 d) 0

6) The inverse Matrix 1 0 0

a 1 0 is

b c 1

a) 1 0 0 b) 1 0 0

-a 1 0 -a 1 0

ac-b -c 1 -b -c 1

c) 1 -a ac- b d) 1 0 0

0 1 -c -a 1 0

0 0 1 ac b 1 7)

7) if A=|aᵢⱼ| and Aᵢⱼ denotes the co-factors of aᵢⱼ , then which of the following is not equal to zero?

a) a₃₁A₁₁+ a₃₂A₁₂ + a₃₃A₁₃

b) a₁₁A₃₁ + a₁₂A₃₂+ a₁₃A₃₃

c) a₂₁A₂₁ + a₂₂A₂₂+ a₂₃A₂₃

d) a₃₁A₂₁ + a₃₂A₂₂+ a₃₃A₂₃

8) The minor of (-4) and 9 and the co-factor of (-4) and 9 in -1 -2 3

-4 -5 -6

-7 8 8 are

a) 42,3; -42,3. b) -42,-3; 42,-3

c) 42,3; -42,-3 d) 42,3; 42,3

9) If A= 0 3 & kA= 0 4a

4 5 3b 60 then the value of k, a and b are respectively ---

a) 12,9,6 b) 9,12,6 c) 12,9,12 c) 16,12,9

10) If the Matrix A satisfies the equation BA = C as B= 1 3 C= 1 1

0 1 0 -1 then which of the following represents A ?

a) 1 4 b) 1 4 c) 1 -4 d) 1 -2

-1 0 0 -1 1 0 0 -1

Mg. A- R.4

1) For any Matrix A, if A⁻¹ exists then which of the following is not true?

a) (A⁻¹)⁻¹= A b) (A')⁻¹ = (A⁻¹)'

c) (A²)⁻¹= (A⁻¹)² d) |A⁻¹|= |A|⁻¹

2) If I is the unit Matrix of order 10x10, then the determinant of I is equal to --

a) 10 b) 1/10 c) 9 d) 1

3) If A and B are two square matrices of the same order, then (A+ B)² is

a) A²- 2AB + B² b) A²- AB - BA + B²

c) A²- 2BA + B² d) A²+ 2AB + B²

4) For how many values of x in the closed interval [-4,-1], the Matrix

3 -1+x 2

3 -1 x+2

x+3 -1 2 is singular?

a) 0 b) 1 c) 2 d) 3

5) If A= 7 1 2 & B= 3 & C= 4

9 2 1 4 2

Find the value AB+ C.

a) 43 b) 43 c) 45 d) 44

44 45 44 45

6) If A= 3 4

5 7 then the value of A(adj A) is equal to

a) I b) |A| c) |A|. I d) none

7) If x+ y 2x + z = 4 7

x - y 2z+ w 0 10 then the values of x, y and z.

a) 2,3,1,2 b) 2,2,3,4 c) 3,3,0,1 d) 2,2,4,3

8) The Matrix 2 k -4

-1 3 4

1 -2 -3 is non-singular if

a) k≠ 2 b) k≠ 3 c) k≠ -3 d) k ≠ -2

9) If A is an invertible Matrix, then the value of det(A⁻¹) is

a) 0 b) I 1/det A d) det A

10) If A= 3 2 & AC = 19 24

4 5 37 46 then Matrix C is:

a) 3 4 b) 3 5 c) 5 4 d) 3 2

5 6 4 6 2 6 6 4

Mg. A - R.5

1) Let A=[5] be a Matrix of order 1x1. Then adjA is equal to

a) [1] b) [5] c) [0] d) 1

2) If A² - A + I =0 then the inverse of A is

a) A - I b) A + I c) A d) I - A

3) If A= 1 1

1 1 then Aⁿ for all n ∈ℕ is:

a) 2ⁿA b) 2ⁿ⁻¹A c) 2ⁿ⁺¹A d) nA

4) Let A and B are two matrices such that AB= A and BA = B. Then A² is equal to -

a) O b) I c) A d) B

5) If A= 4 2

-1 1 then the value of (A - 2I)(A - 3) is

a) A b) I c) O d) 4I

4) If A= 1 -2 & B= a 1

2 -1 b -1 and (A+ B)² = A²+ B², then the values of a and b are

a) 4,1 b) 1,4 c) 0, 4 d) 2, 4

5) Let A= a 0 & B= 1 0

1 1 5 1 if A² = B, then the value of a is.

a) 1 b) -1 c) 4 d) no real value of a

6) The matrix 0 7 4

-7 0 -5

-4 5 0 is

a) symmetric b) skew-symmetric c) non singular d) orthogonal

7) Let A= 1 -1 1 4 2 2

2 1 3 & 10B = -5 0 a

1 1 1 1 -2 3 if B is the inverse of the Matrix A, then the value of a

a) 2 b) -1 c) -2 d) 5

8) If A= 0 0 -1

0 -1 0

-1 0 0 then the only correct statement about the Matrix A is --

a) A⁻¹does not exist b) A= (-1)I

c) A is a zero Matrix. d) A² = I

9) If A= -1 2 & B= 3 -2

3 4 1 5 then find a Matrix C such that 2A +B +C is the zero Matrix of order 2.

a) -1 -2 b) 1 -2 c) -1 -2 d) 1 2

-7 -13 -7 13 7 13 7 13

10) If 2A+ B= 2 3 & 3B - 2A = 10 1

5 1 3 5 Find the Matrix A and B.

a) -1/2 2 & 3 1

3/2 -1/4 2 3/2

b) -1/2 2 & 3 -1

3/2 -1/4 2 3/2

c) 1/2 2 & -3 1

3/2 -1/4 2 3/2

d) 1/2 2 & 3 1

3/2 -1/4 2 -3/2

Mg. A- R.6

1) If A is and m x n matrix and B is n x p matrix does AB exists? If yes, write its order.

a) Yes,.mxp b) yes, mx n c) yes,mx q d) n

2) If A= 2 1 4 and B= 3 -1

4 1 5 2 2

1 3 , write the order of AB and BA.

a) 2x2, 3x3 b) 2x3, 3x2 c) 2x4, 3x3 d) 1x1, 2x2

3) If A= 4 3 and B= - 4

1 2b 3 Find AB.

a) 7 b) -7 c) 2 d) -2

2 -2 7 -7

4) If A= 2 -1

-1 2 and I is the unit Matrix of order 2, then A² is

a) 4A - 3I b) 3A - 4I c) A- I d) A+ I

5) Find the multiplicative inverse of 2 1

7 4

a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1

-7 -2 -7 2 7 2 7 -2

6) If A= 1

3 write AA'.

a) 1 2 3 b) 1 -2 3 c) -1 2 3 d) -1 -2 -3

2 4 6 2 4 6 2 4 6 2 4 6

3 6 9 3 6 9 3 6 9 3 6 9

7) A is a square Matrix such that A³= I; then inverse of A is

a) A² b) A c) A³ d) none

8) Assuming that the sum and products given below are defined, which of the following is not true for Matrices?

a) AB= AC done not imply

b) A+ B = B+ A

c) (AB)' = B' A'

d) AB = O implies A= O or, B= O

9) Give an example of two non-zero 2 x 2 matrices A and B such that AB= O.

10) If A= 2 3

5 7 find A+ A'.

a) 4 8 b) -4 8 c) 4 -8 d) -4 -8

8 1 8 4 8 14 14 8

Mg. A- R.8

1) If A= cosx sinx

- sinx cosx find x satisfying 0< x < π/2 when A+ A'= I

a) π b) π/2 c) π/3 d) π/4

2) If A=-1 2 & B= 3 -2

3 4 1 5 then find a Matrix C such that 2A + B+ C is the zero Matrix of order 2.

a) -1 -2 b) 1 -2 c) -1 2 d) 1 2

-7 -13 7 13 7 13 7 13

3) Determine two matrices A and B, when

2A+ B= 2 3 & 3B -2A= 10 1

5 1 3 5

3) If A= cosx -sinx

sinx cosx. find AA'.

a) 0 1 b) 1 0 c) 0 0 d) 1 1

0 1 0 0 1 1 1 0

4) If A= 1 0 and B= x 0

y 5 1 -2 and the relation A+ 2B= I, where I is 2 x 2 unit matrix. find x and y.

a) 0,1 b) 0,-1 c) 0,2 d) 0,-2

5) If A=1 2 B=1 2

1 2 3 4 determine AB, BA. 4 6 3 5

7 10 7 11

6) If A= cosx sinx & B= cosy siny

sinx cosx siny cosy show that AB= BA

7) If A= 3 2 5 & B= 1 2 & C= 7 -8

7 -4 0 2 -1 5 9

3 5

Find the value of AB- C. 15 37

-6 9

8) If A= 2 5 & X= x and B= 5

3 4 y 8 then show from the relation A.X = B that 2x+ 5y = 5 and 3x + 4y= 8.

9) If A= 1 -1

-1 1 satisfies the matrix equation A²=kA,write the value of k.

a) 0 b) 1 c) 2 d) 3 e) none

10) If B= 1 -2 & B= 6 0

1 4 0 6 find A if the relation A.B= C. 4 2

-1 1

Mg. A-R.9

1) If A= 2 -3 & B= 1 5 b & C= 2 4 1

1 a 0 2 -3 1 -1 5 with the relation AB= C then find a,b

a) 3, 4 b) -3, -4 c) -3, 4 d) 3-, 4

2) If A=2 3 & B= 1 0

3 10 0 1 then find (2I - A)(10I - A), Where I is identity Matrix.

a) 9 b) I c) 3 d) 9I

3) If A= 1 1

1 1 satisfies A⁴= kA, then write the value of k.

a) 0 b) 1 c) 2 d) -1 e) none

4). -1 0 0

If A= 0 -1 0

0 0 -1 find A².

5) If A= x 2 and B= 3

4 and the relation AB= 2 find x.

a)1 b) 2 c) -2 d) none

6) If A= 2 3 & B= 3 7

6 5 4 0 then find (A+ B)². 125 100

100 125

7) If A= [aᵢⱼ] is a 2 x 2 matrix such that aᵢⱼ= i+ 2j, write A.

8) If A= 2 -2 2 0

-3 4 find (-A²+6A). 0 2

9) If A= 1 0 6 0

2 1 then 2A²+4I is . 8 6

10) write matrix A satisfying A+ B= C where B= 2 3 & C = 3 -6

-1 4 -3 8

Mg. A- R.10

1) If f(x)= x²+ 2x and A= 1 2

4 -3 find f(A)

11 0

0 11

2) if A= [aᵢⱼ] is a square Matrix that aᵢⱼ = i² - j², then write whether A is symmetric or skew symmetric.

3) If A= 1 3

3 4 then A²- 5A - 2I is

a) 0 b) 1 c) -1 d) 2

4) If A= 3 -2 and B= 1 0

4 -2 0 1 then find the value of k such that, A²= kA - 2I.

a) 0 b) 1 c) -1 d) -2

5) For any square matrix write whether AA' is symmetric or skew-symmetric.

6) If A= 1 0 1

0 0 0

1 0 1 then A² is

a) A b) A' c) 2A d) 2A'

7)If A= 3 2 2 -1

5 4 then inverse of A is: -5/2 3/2

8) If A=2 3

5 -2 then inverse of A is:

a) A b) A' c) A/91 c) A/19

9) If A= [aᵢⱼ] is a skew symmetric matrix, then write the value of ᵢ∑aᵢⱼ

10) If A= [aᵢⱼ] is a skew symmetric matrix, then write the value of ᵢ∑ ⱼ∑aᵢⱼ .

Mg. A-R.11

1) If A and B are symmetric matrices, then write the condition for which AB is also symmetric.

2) If A= 3 5

7 -11 then AA⁻¹ is:

a) A b) A⁻¹ c) 0 d) I

3) If A= 3 5

7 -11 then A⁻¹A is:

a) A b) A⁻¹ c) 0 d) I

4) If B is a skew-symmetric matrix, write whether the matrix AB A' is symmetric or Skew-symmetric.

5) Find the values of x and y, if 2A+ B= C where A= 1 3 B= y 0 & C= 5 6

0 x 1 2 1 8

6) If x+ 3 4 = 5 4

y -4 x+ y 3 9 then find x and y.

7) If A is a singular Matrix, then write the value of |A|.

8) For what value of x, the following matrix is singular ?

5 - x x+1

2 4

9) write the value of the determinant of

2 3 4

2x 3x 4x

5 6 8

10) State whether the matrix 2 3

6 4 is singular or non-singular.

Mg. A- R.12

1) find the value of the determinant of

4200 4201

4202 4203

2) find the value of the determinant

101 102 103

104 105 106

107 108 109

3) Write the value of the determinant

a 1 b+c

b 1 c+a

c 1 a+b

4) If A= 0 i & B= 0 1

i 1 1 0 find the value of |A|+| B|.

5) if A= 1 2 & B= 1 0

3 -1 -1 0 find |AB|.

6) If A= 1 2 and B= 1 -4

3 -1 3 -2 find |AB|.

7) If A=[aᵢⱼ] is a 3x3 diagonal matrix such that a₁₁=11, a₂₂= 2 and a₃₃ = 3, then find |A|.

8) If A=[aᵢⱼ] is a 3x3 scalar Matrix such that a₁₁=2, then write the value of |A|.

9) If I₃ denotes Identity matrix of order 3x3, write the value of its determinant.

10) A matrix A of order 3x3 has determinant 5. What is the value of |3A| ?

Mg. A- R.13

1) On spending by first row, the value of the determinants of 3x3 square Matrix A=[aᵢⱼ] is a₁₁ C₁₁+ a₁₂C₁₂+ a₁₃ C₁₃, where Cᵢⱼ is the cofactor of aᵢⱼ in A. write the expression for its value on expanding by the second column.

2) let A= [aᵢⱼ] be a square matrix of order 3x3 and Cᵢⱼ denote cofactor of aᵢⱼ in A. If |A|= 5, write the value of a₃₁C₃₁+ a₃₂C₃₂+ a₃₃Ca₃₃.

3) In question 18, write the value of write the value of a₁₁C₂₁+a₁₂C₂₂ + a₁₃ C₂₃ .

4) Write the value of

sin 20 - cos 20

sin 70 cos 70

5) If a square Matrix satisfying A'. A = I, write the value of |A|.

6) If A and B are square matrices of the same order such that |A| = 3 and AB= I, then write the value |B|.

7) A is a skew-symmetric of order 3, write the value of |A|.

8) If A is a square matrix of order 3 with the determinants 4, then write the value of |-A|.

9) if A is square Matrix such that |A|= 2, Write the value of |AA'|.

10) find the value of the determinant

243 156 300

81 52 100

-3 0 4

Mg. A- R.14

1) write the value of the determinant of

2 -3 5

4 -6 10

6 -9 15

2) If the matrix 5x 2

-10 1 is singular, find the value of x.

3) If A is a square matrix of order n x n such that |A| = K, then write the value of |-A|.

4) find the value of the determinant

2² 2³ 2⁴

2³ 2⁴ 2⁵

2⁴ 2⁵ 2⁶

5) if A and B are nonsingular matrices of the same order, write whether AB is singular or non-singular.

6) A metrix of order 3 x 3 has determinant 2. What is the value of |A(3I)|, where I is the identity matrix of order 3 x 3.

7) If A and B are square matrices of order 3 such that |A| = -1, |B| = 3, then find the value of |3AB|

8) write the cofactor of a₁₂ in the following metrix 2 -3 5

6 0 3

1 5 -7

9) If 2x+5 3

5x+2 9 = 0, find x.

10) Write the adjoint of the matrix

A= -3 4

7 -2

Mg. A- R.15

1) If A is square matrix such that A(adj A)= 5I, where I denotes the identity matrix of the same order. Then, find the value of |A|.

2) If A is a square Matrix of order 3 such that |A|= 5, write the value of |adj A|.

3) If A is a square matrix of order 3 such that |adj A|= 64, find |A|.

4) if A is a non-singular Square matrix such that |A|=10, find |A⁻¹|

5) If A, B, C are three non-null square matrices of the same order, write the condition on A such that AB= AC => B= C.

6) If A is a non-singular square Matrix such that A⁻¹= 5 3

-2 -1 then find (A')⁻¹.

7) if adj A= 2 3

4 -1 and adjoint of

B= 1 -2

-3 1 find adj of AB.

8) If A is a symmetric matrix, write whether A' is symmetric or Skew-symmetric.

9) If A is a square matrix of order 3 such that |A| = 2, then write the value of adj(adj A).

10) If A is a square matrix of order 3 such |A|=3, then find the value of |adj (adj A)|.

Mg. A- R.16

80) If A is a square matrix of order 3 such that adj(2A)= k adj(A), then write the value of k.

81) If A is a square matrix, then write the matrix adj(A') - (adj)'.

82) Let A be a 3 x 3 square Matrix such that A(adj A)= 2I, where I is the identity Matrix, write the value of |adj A|.

83) if A is a nonsingular symmetric matrix, write whether A⁻¹ is Symmetric or Skew-symmetric.

84) If A= cosx Sinx

- sinx cos x and A(adjoint A)= k 0

0 k then find the value of k.

85) If A is an invertible Matrix such that |A⁻¹|= 2, find the value of |A|

86) If A is a square Matrix such that A(adj A)= 5 0 0

0 5 0

0 0 5 then write the value of |adj A|.

87) If A= 2 3

5 -2 be such that A⁻¹= kA, then find the value of k.

88) Let A be a square matrix such that A² - A + I= O, then write A⁻¹ in terms of A.

89) using cramer's rule write the solution of the system of equations 3x+4y=7; 7x - y= 6.

90) find the inverse of the matrix

3 -2

-7 5

91) find the inverse of cosx sinx

-sinx cosx

92) If A= 1 -3

2 0 write adj A.

93) If A= a b & B = 1 0

c d 0 1 find adj(AB).

94) If A= 1 0 0 & B= x & C= 1

0 1 0 y -1

0 0 1 z 0 with the relation AB= C, then find x,y, z.

95)If A= 1 0 0 & B= x & C= 1

0 -1 0 y 0

0 0 -1 z 0 with the relation AB= C, then find x,y and z.

96)If A= 1 0 0 & B= x & C= 1

0 y 0 -1 0

0 0 1 z 1 with the relation AB= C, then find x,y and z.

97) If A= 3 -4 0 & B= x

9 2 0 y with the relation AB= C, then find x,y.

98) If A= 1 0 0 & B= x & C= 2

0 0 1 y -1

0 1 0 z 3 with the relation AB= C, then find x,y, z.

99) If A= 2 4 & B= n & C= 8

4 3 1 11 with the relation AC= B, then find n.

MULTIPLE CHOICE QUESTIONS

----------*******------------*********--------

1) If A= 1 0 0

0 1 0

a b -1 then A² is equal to

A) null matrix B) a unit matrix.

C) - A D) A

2) If A= i 0

0 i, n∈ ℕ, then A⁴ⁿ equal

A) 0 i B) 0 0 C). 1 0 D) 0 i

i 0 0 0 0 1 i 0

3) If A and B are two Matrices such that AB= A and BA= B, then B² is equals to

A) B. B) A C) 1 D) 0

4) If AB= A and BA= B, where A and B are square matrices, then

A) B²= B and A²= A.

B) B²≠ B and A²= A

C) A²≠ A and B²= B

D) A²≠ A and B²≠ B

5) If A and B are two Matrices such that AB= B and BA= A, then A²+ B² is equal to

A) 2AB B) 2BA C) A+B. D) AB

7) if the matrix AB is zero, then

A) it is not necessary that either A= O or B= O.

B) A= O or B= O C) A= O and B= O

D) all the above Statements are wrong.

9) If A and B are square matrices of order 3, A is non-singular and AB= O, then B is a

A) null matrix. B) singular Matrix

C) unit Matrix

D) non singular matrix

10) If A= n 0 0 and B= a₁ a₂ a₃

0 n 0 b₁ b₂ b₃

0 0 n c₁ c₂ c₃ then AB is equals to

A) B B) nB. C) Bⁿ D) A+ B

11) If A= 1 a

0 1 then Aⁿ (where n ∈ ℕ) equals

A) 1 na. B) 1 n²a C) 1 na D) n na

0 1 0 1 0 0 0 n

12) If A= 1 2 x and B= 1 -2 y

0 1 0 0 1 0

0 0 1 0 0 1 and AB= I, then x+ y is

A) 0. B) -1 C) 2 D) none

13) If A= 1 -1 & B=a 1

2 -1 b -1 and (A+ B)² = A²+ B², values of a and b

A) a= 4, b= 1 B) a= 1, b= 4.

C) a= 0, b= 4 D) a= 2, b= 4

14) If A= a b

c -a is such that A²= I, then

A) I+ a² + bc= 0 B) I- a² + bc= 0

C) I- a² - bc= 0. D) I+ a² - bc= 0

15) If S= [Sᵢⱼ] is a scalar matrix such that sᵢⱼ = k and A is a square matrix of the same order, then AS= SA= ?

A) Aᵏ B) k+ A C) kA. D) kS

16) If A is a square Matrix such that A² = A, then (I+A)³ - 7A is equals to

A) A B) I - A C) I. D) 3A

17) if a matrix A is both symmetric and skew symmetric, then

A) A is a diagonal matrix

B) A is zero matrix.

C) A is scalar matrix

D) A is a square matrix

18) the matrix 0 5 -7

-5 0 11

7 -11 0 is

A) a skew-symmetric matrix

B) a symmetric matrix

C) a diagonal matrix.

D) an upper triangular matrix

19) If a square matrix, then AA is a

A) skew symmetric matrix

B) symmetric matrix

C) diagonal Matrix

D) none.

20) If A and B are symmetric matrices, then ABA is

A) symmetric matrix.

B) skew-symmetric matrix

C) diagonal Matrix

D) scalar matrix

21) If A= 5 x

y 0 and A= A', then

A) x= 0, y= 5 B) x+y= 5

C) x= y. D) none

22) If A= 3 x 4 matrix and B is a matrix such that A'B and BA' are both defined. then, B is of the type

A) 3x4. B) 3x3 C) 4x4 D) 4x3

23) If A= [aᵢⱼ] is a square matrix of even order such that aᵢⱼ = i² - j³, then

A) A is a skew-symmetric matrix and |A|= 0

B) A is symmetric metrix and |A| is a square.

C) A is a symmetric Matrix and |A|= 0 D) none.

24) If cosx - sinx

sin x cosx then A'+ A= I, if

A) x=nπ, n ∈ Z

B)x=(2n+1)π/2 , n ∈ Z

C) x= 2nπ +π/3, n ∈ Z. D) none

25) If A= 2 0 -3

4 3 1

-5 7 2 is expressed as the sum of a symmetric and skew symmetric matrix, then the symmetric matrix is

A) 2 2 -4 B) 2 4 -5

2 3 4 0 3 7

-4 4 2. -3 1 2

C) 4 4 -8 C) 1 0 0

4 6 8 0 1 0

-8 8 4 0 0 1

26) Iut of the given matrices, choose that matrix which is a scalar Matrix:

A) 0 0 B) 0 0 0 C) 0 0 D) 0

0 0. 0 0 0 0 0 0

27) the number of all possible matrices of order 3x3 with entry 0 or 1 is

A) 27 B) 18 C) 81 D) 512

28) Which of the given values of x and y make the following pairs of matrices equal ?3x+7 5 = 0 y-2

y+1 2 - 3x 8 4

A) x= -1/3, y= 7

B) x= 7, y= 2/3

C) x= -1/3, y= -2/5

D) not possible to find.

29) If A= 0 2 and kA= 0 3a

3 -4 2b 24

then the values of k, a, b are respectively

A) -6,-12,-18 B) -6,4,9

C) -6,-4,-9. D) -6,12,18

30) If I= 1 0 and J= 0 1

0 1 -1 0 and B= cosx sinx

-sinx cosx then B equals

A) I cosx + J sinx.

B) I sinx + J cosx

C) I cosx - J sinx

D) - I cosx + J sinx

31) The trace of the matrix

A= 1 -5 7

0 7 9

11 8 9 is

A) 17. B) 25 C) 3 D) 12

32) If A=[aᵢⱼ] is a scalar matrix of order n x n such that aᵢⱼ = k, for all i, then trace of A equal to

A) nk. B) n+ k C) n/k D) none

33) If A and B are square matrices of order 2, then det(A+B)= 0 is possible only when

A) det(A+B)= 0 or det(B)= 0

B) det(A)= 0 + det(B)= 0

C) det(A)= 0 and det(B)= 0

D) A'+ B= O.

34) which of the following is not correct in a given determinant of A, where A=[aᵢⱼ]₃ ₓ ₃.

A) Order of minor is less than order of the det (A)

B) minor of an element can never be equal to cofactor of the same element.

C) Value of a is determinant is obtained by multiplying elements of a row or column by corresponding cofactors

D) order of minors and cofactors of the elements of A is same

35) Let x 2 x

x² x 6

x x 6

Then the value ofax⁴+bx³+cx²+dx+e is equal to

A) 0 B) -16 C) 16 D) none.

36) the value of the determinant. a² a 1

cos nx cos(n+1)x cos(n+2)x

sin nx sin(n+1)x sin(n+2)x is independent of

A) n. B) a C) x D) none

37) If ∆₁= 1 1 1 & ∆₂= 1 bc a

a b c 1 ca b

a² b² c² 1 ab c then A) ∆₁ +∆₂= 0. B) ∆₁+2∆ = 0 C) ∆₁+ ∆₂ D) none

38) Dₖ = 1 n n

2k n²+n+2 n²+n

2k-1 n² n²+n+2 and ⁿₖ₌₁∑ Dₖ = 48, then n equals

A) 4. B) 6 C) 8 D) none

39) Let x²+ 3x x-1 x+3

x+1 -2x x-4

x-3 x+4 3x

= ax⁴ + bx³+ cx² + dx+ e be an identity in x, where a, b, c, d, e are independent of x. Then the value of e is

A) 4 B) 0. C) 1 D) none

40) using the factor theorem it is found that a+b, b+c and c+a are three factors of the determinant

-2a a+ b a+ c

b+a -2b b+c

c+a c+b -2c The other factor in the value of the determinant is

A) 4. B) 2 C) a+b+c D) none

41) if a,b,c are distinct, then the value of x satisfying

0 x²-a x³- b

x²+a 0 x²+ c = 0 is

x⁴+ b x- c 0

A) c B) a C) b D) 0

42)

43)

44)

45)

46) The value of 5² 5³ 5⁴

5³ 5⁴ 5⁵

5⁴ 5⁵ 5⁶ is

A) 5² B) 0 . C) 5¹³ D) 5⁹

47)

48)

49)

50) If A is an invertible matrix, then which of the following is not true

A) (A²)⁻¹= (A⁻¹)² B) |A⁻¹|= |A⁻¹|.

C) (A')⁻¹= (A⁻¹)' D) |A| ≠ 0

51) If A is an invertible matrix of order 3, then which of the following is not true

A) |adj A|= |A|² B) (A⁻¹)⁻¹= A

C) If BA= CA, then B= C, where B and C are square matrices of order 3.

D) (AB)⁻¹= B⁻¹A⁻¹, where B= [bᵢⱼ]₃ ₓ ₃ and |B| ≠ 0

52) If A= 3 4 & B= -2 -2

2 4 0 -1 then (A+B)⁻¹=

A) is a skew-symmetric matrix

B) A⁻¹+ B⁻¹ C) does not exist D) n.

53) If S= a b

c d then adj A is

A) -d -b B)d -b C) d b D) d c

-c a -c a. c a b a

54) If A is a singular matrix, then adj A is

A) non-singular B) singular.

C) symmetric D) not defined

55) If A, B are two n x n non-singular matrices, then

A) AB is non-singular.

B) AB is singular

C) (AB)⁻¹= A⁻¹B⁻¹

D) (AB)⁻¹ does not exist

56) If A= a 0 0

0 a 0

0 0 a then the value of |adj A| is

A) a²⁷ B) a⁹ C) a⁶. D) a²

57) 1 2 -1

If A= -1 1 2

2 -1 1 then det(adj(adj A)) is

A) 14⁴. B) 14³ C) 14² D) 14

58) If B is a non-singular matrix and A is a square matrix, then det(B⁻¹ AB) is equal to

A) Det (A⁻¹) B) Det (B⁻¹)

C) Det (A). D) Det (B)

59) For any 2 x 2 matrix, if A(adj A) = 10 0

0 10 then |A| is equal to

A) 20 B) 100 C) 10. D) 0

60) If A⁵ = O such that Aⁿ for 1≤n ≤ 4, then (I - A)⁻¹ equals

A) A⁴ B) A³ C) I+ A D) none.

61) If A satisfies the equation x³ - 5x³ + 4x + K= 0, then A⁻¹exists if

A) K≠ 1 B) K≠2 C) K≠-1 D) none.

62) If for the matrix A, A³ = I, then A⁻¹=

A) A². B) A³ C) A D) none

63) If A and B are square matrices such that B= - A⁻¹BA, then (A+ B)²

A) O. B) A²+ B². C) A²+2AB+B² D) A+B

64) 2 0 0

0 2 0

0 0 2 Then A⁵ is

A) 5A B) 10A C) 16A. D) 32A

65) For non-singular square matrix A, B and C of the same order (AB⁻¹C)=

A) A⁻¹BC⁻¹ B) C⁻¹B⁻¹A⁻¹

C) CBA⁻¹ D) C⁻¹BA⁻¹.

66) 5 10 3

the matrix -2 -4 6

-1 -2 b is a singular matrix, if the value of b=

A) -3 B) 3 C) 0 D) non-existent.

67) If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is

A) dⁿ B) dⁿ⁻¹. C) dⁿ⁺¹ D) d

68) if A is a matrix of order 3 and |A| = 8, then |adj A|=

A) 1 B) 2 C) 2³ D) 2⁶.

69) If A² - A + I= 0, then the inverse of A is.

A) A⁻² B) A+ I C) I - A. D) A - I

70) if A and B are invertible matrices, which of the following statement is not a correct.

A) adj A= |A| A⁻¹

B) det(A⁻¹)= (A)⁻¹

C) (A+B)⁻¹ = A⁻¹+ B⁻¹.

D) (AB)⁻¹ = B⁻¹A⁻¹

71) If A is a square Matrix such that A² = I, then A⁻¹ is equal to

A) A+ I B) A. C) 0 D) 2A

72) Let A= 1 2 & B= 1 0

3 -5 0 2 and X be a matrix such that A= BX, then X is equal to

A) 1 2 B) -1 2 C) 2 4

3 -5. 3 5 3 -5 D) n

73) If A= 2 3

5 -2 be such that A⁻¹ = kA, then k equal to

A) 19 B) 1/19. C) -19 D) -1/19

74) If A= 1/3 1/3 2/3

2 1 -2

x 2 y is orthogonal, then x+y=

A) 3 B) 0 C) -3. D) 1

75) if A= 1 0 1

0 0 1

a b 2 then aI + bA + 2A² is

A) A B) -A C) ab A D) none .

76) If A= 1 - tan x & B= 1 tanx

tanx 1 -tanx 1 and C= a - b

b a and relation of A B⁻¹= C then

A) a=1, b= 1

B) a=cos 2x , b= sin 2x.

C) a=sin 2x, b= cos 2x D) n

77) If a matrix A is such that 3A³ + 2A² + 5A + I= 0, then A⁻¹ is

A) -(3A² + 2A + 5)

B) (3A² + 2A + 5)

C) (3A² - 2A - 5) D) none .

78) If A is an invertible Matrix, then det A⁻¹ is equal to

A) det(A) B) 1/det(A). C)1 D) n

79) If A= 2 -1

3 -2 then Aⁿ=

A) A= 1 0

0 1 if n is an even natural number.

B) A= 1 0

0 1 if n is an odd natural number

C) A= -1 0

0 1 if n belongs to N

D) none

If B is a symmetric matrix, write whether the matrix AB A' is symmetric or Skew-symmetric.

26) If A is a skew-symmetric and n ∈ ℕ such that (Aⁿ)ᵀ = K Aⁿ, write the value of K.

27) If A is a symmetric matrix and n∈ ℕ, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two.

28) if A is a symmetric matrix and n is an even natural number, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two.

29) If A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two.

30) If A and B are symmetric matrices of the same order, write whether AB - BA is symmetric or Skew-symmetric or neither of these two.

31) write a square matrix which is both symmetric as well as skew symmetric.