EXERCISE - 1
1) A matrix A= [aᵢⱼ] is an upper triangular matrix if.
A) it is a square Matrix and aᵢⱼ = 0, i< j
B) it is a square Matrix and aᵢⱼ = 0, i> j.
C) it is not a square Matrix and aᵢⱼ = 0, i>j
D) it is not a square Matrix and aᵢⱼ = 0, i< j
2) if A is any mxn Matrix such that AB and BA are both, defined, then B is an
A) mxn matrix B) n x m matrix.
C) n x n matrix D) mxm matrix
3) If E(x)= cosx sinx
- sinx cosx then E(y) E(z) is equal to
A)E(0') B)E(yz)
C) E(y+z). D) E(y-z)
4) If E(x)= cos²x cosx sinx
cosx sinx sin²x
and x and y differ by an odd multiple of π/2, then E(x) E(y) is a
A) null matrix. B) unit Matrix
C) diagonal Matrix D) none
5) If If A= cos²x cosx sinx
cosx sinx sin²x
And. B = cos²y cosy siny
cosy sinx sin²y
are two matrices such that the product AB is the null metrix, than x - y is...
A) 0 B) multiple of π
C) an odd multiple of π/2. D) none
6) The matrix A satisfying the equation 1 3 X A= 1 1
0 1 0 -1 is..
A) 1 4 B) 1 -4 C) 1 4 D) none
-1 0 1 0 0 -1.
7) If I= 1 0 & J= 0 1 B= cosx sinx
0 1 -1 0 -sinx cosx then B equal to
A) I cosx + J sinx.
B) I sinx+J cosx
C) I cosx - J sinx
D) - I sinx+J cos x
8) If A is a square Matrix such that A A' = I = A' A, then A is
A) a symmetric matrix
B) A skew symmetric matrix.
C) A diagonal Matrix
D) an orthogonal matrix.
9) If A is an orthogonal Matrix, then A⁻¹ equals
A) A B) A' . C) A² D) none
10) If D= diag(d₁ , d₂, d₃, ...dₙ) , where dᵢ ≠ 0 for all i= 1,2,....,n, then D⁻¹ is equal to
A) D B) diag(d₁⁻¹d₂⁻¹,..dₙ⁻¹).
C) Iₙ D) none
11) if A and B are two invertible matrices, then the Inverse of AB is equal to..
A) AB B) BA C) A⁻¹B⁻¹ D) B⁻¹A⁻¹.
12) If A, B, C are invertible matrices, then (ABC)⁻¹ is equal to..
A) A⁻¹B⁻¹C⁻¹. B) B⁻¹C⁻¹A⁻¹
C) C⁻¹A⁻¹B⁻¹ D) C⁻¹B⁻¹A⁻¹
13) If A= diag(d₁ , d₂, d₃, ...dₙ), then Aⁿ is equal to...
A) diag ( d₁ⁿ⁻¹,d₂ⁿ⁻¹,d₃ⁿ⁻¹,dₙⁿ⁻¹)
B) d₁ⁿ,d₂ⁿ,d₃ⁿ,....dₙⁿ). C) A D) n
14) If A and B are two square matrix such that then AB= A and BA= B, then
A) A,B are indempotent.
B) only A is indempotent
C) only B is idempotent D) n
15) The inverse of a symmetric matrix is
A) symmetric.B) skew-symmetric
C) diagonal matrix D) none
16) The inverse of a diagonal matrix is .
A) A symmetric matrix
B) a skew-symmetric matrix
C) A diagonal matrix. D) none
17) if A is symmetric matrix and n belongs to N then Aⁿ is...
A) symmetric.
B) skew-symmetric
C) a diagonal matrix D) none
18) If A is a skew-symmetric matrix and n is a positive integer, then Aⁿ is..
A) a symmetric matrix
B) A skew-symmetric matrix
C) diagonal matrix D) none.
19) if A is a skew-symmetric matrix and n is odd positive integer, then Aⁿ is...
A) a symmetric matrix
B) a skew-symmetric matrix.
C) a diagonal Matrix D) none
20) If A is a skew-symmetric matrix and n is an even positive integer, then Aⁿ is...
A) a symmetric matrix.
B) A skew symmetric matrix
C) A diagonal Matrix D) none
21) If A= [aᵢⱼ] is a skew-symmetric matrix of order n, then aᵢⱼ =
A) 0 for some i
B) 0 4 all i=1,2..j.
C) 1 for some i
D) 1 for all I= 1,2.....n
22) if A, B are symmetric matrices of the same order then AB - BA is
A) symmetric matrix
B) skew-symmetric matrix.
C) null matrix D) unit Matrix
23) if A and B are matrices of the same order, then (A+B)²= A²+ 2AB + B² is possible, if ...
A) AB= I B) BA= I C) AB= BA. D) N
24) 1 -5 -7
If A= 0 7 9
11 8 9
then, trace of matrix A is
A) 17. B) 25 C) 3 D) 12
25) If A is a skew symmetric matrix, then trace of A is
A) 1 B) - 1. C) 0 D) none
26) If A, B are square matrices of the same order, then adj(AB) is equal to
A) (adj A)(adj B) B) (adj B)(adj A).
C) adj A +adj B D) adj A - adj B
27) If A is a square matrix of order n xn and k is a scalar, then adj(k A) is equals to.
A) k adj A B) kⁿ adj A
C) kⁿ⁻¹ adj A. D) kⁿ⁺¹ adj A
28) If A is a square matrix, then A adj A is equals to
A) A B) A⁻¹ C) I D) |A| I.
29) If is a square matrix or order n x n, then adj (adj A) is equals to..
A) |A|ⁿ A B) |A|ⁿ⁻¹A
C) |A|ⁿ⁻²A . D) |A|ⁿ⁻³A
30) If A is a singular Matrix, then A adj A is
A) identity Matrix B) null matrix.
C) scalar Matrix D) none
31) If A= cosx sinx
- sinx cosx and
A adj A = k. 1 0
0 1 then the value of k is..
A) sinx cosx B) 1. C) 2 D) 3
32) If A= 1 1
1 1 and n belongs to N, then Aⁿ is equals to..
A) 2ⁿ A B) 2ⁿ⁻¹ A. C) nA D) none
33) If A= a b & A²= x y
b a y x then
A) x= a²+ b² B) x= a²+b², y= 2ab.
C) x= a²+b², y= a² - b²
D) x = 2ab, y= a² + b²
34) If A is a square Matrix, then adj A' - (adj A)' is equal to
A) 2|A| B) 2|A| I
C) null metric. D) unit Matrix
35) IF A= 1 3
3 4 and A² - kA - 5= O, then the value of k is
A) 3 B) 5. C) 7 D) - 7
36) If A= [aᵢⱼ] is a scalar Matrix, then trace of A is..
A) ᵢ∑ⱼ ᵢ∑ aᵢⱼ B) ⱼ∑ aᵢⱼ
C) ⱼ ∑aᵢⱼ D) ᵢ ∑aᵢⱼ.
37) if A= [aᵢⱼ] is a scalar matrix of order n x n such that aᵢⱼ = k for all i, then trace of A is equal to
A) nk. B) n+k C) n/k D) none
38) If A= [aᵢⱼ] is a scalar matrix of order n x n such that aᵢⱼ= k for all i, then |A|=
A) nk B) n+k. C) nᵏ D) kⁿ.
39)If A= [aᵢⱼ] is a square matrix of order n x n and k is a scalar then |kA| =
A) kⁿ |A|. B) k|A| C) kⁿ⁻¹|A| D)none
40) cos a - sina 0
Let F(a)= sina cosa 0
0 0 1
Where a belongs to R. Then (F(a))⁻¹ is equal to
A) F(a). B) F(a⁻¹) C) F(2a) D) n
41) cosx - sinx 0
If F(x)= sinx cosx 0
0 0 1
cosx 0 sinx
And G(y)= 0 1 0
- sinx 0 cosx
[F(x) G(y)]⁻¹ is equal to
A) F(-x) G(-y) B) F(x⁻¹) G(y⁻¹)
C) G(-y) F(-x) D) G(y⁻¹)F+(x⁻¹)
42)If A= 2 1 B= 3 2 C= 1 0
3 2 5 -3 0 2 and AMB= C then find metrix M
A) 1 1 B) 1 1 C)1 0 D) 0 1
1 0 0 1 1 1 1 1
43) A= 1 - tanx B= 1 tanx
tanx 1 -tanx 1
C= a - b
b a and A B⁻¹= C then find the value of a and b
A) 1,1 B) cos 2x, sin 2x
C) sin 2x , cos 2x D) none
44) If A and B are Matrices such that AB and A+B both are defined, then
A) A and B can be any two matrices.
B) A and B are square matrices not necessarily of the same order
C) A, B are square matrices of the same order.
D) number of columns of A is same as the number of rows of B
45) if a matrix A is such that 3A³ + 2A² + 5A + I = 0, then A⁻¹ is equal to
A) -(3A² +2A+5) B) 3A²+2A+5
C) 3A² -2A -5 D) NONE
46) A and B be 3 x 3 matrices. Then AB= 0 implies
A) A= 0 and B= 0 B) |A|=0, |B|=0
C) either |A|0 or |B|=0
D) A= 0 or B= 0
47) Which of the following is incorrect.
A) A² - B² = (A+B)(A-B)
B) (A')' = A
C) (AB)ⁿ= AⁿBⁿ where A, B commute
D) (A- I)(A+I)=0 <=> A² = I
48) If is an invertible Matrix, then which of the following is correct.
A) A⁻¹ is multi-valued
B) A⁻¹ is a singular
C) (A⁻¹)' ≠ (A')⁻¹ D) |A|≠ 0
49) which of the following
A) adjoint of symmetric matrix is symmetric
B) adjoint of a matrix is unit matrix
C) A(adjoint A)= (adj A) A= |A| I
D) adjoint of a diagonal Matrix is a diagonal matrix is/are incorrect
a) A b) B c) C and D d) none
50) If a b
c -a is to be the square root of the two-rowed unit Matrix, then a, b and c should satisfy the relation
A) 1+a²+bc= 0 B) 1- a² - bc= 0
C) 1 - a² + bc= 0 D) a²+bc- 1= 0
51) If for a matrix A, A² + I= 0, where I is the identity Matrix, then A equals
A) 1 0 B) -i 0 C)1 2 D) -1 0
0 1 0 -i -1 1 0 -1
52) If A=[aᵢⱼ]ₘₓₙ is a matrix of rank r, then
A) r= min (m,n) B) r< min(m,n)
C) r≤ min (m,n) D) none
53) If Iₙ is the identity matrix of order n, then rank of Iₙ is
A) 1 B n C) 0 D) none
54) If A =[aᵢⱼ]ₘₓₙ is a matrix of rank r and B is a square submatrix of order r+2, then
A) B is invertible
B) B is not invertible
C) B may or may not be invertible
D) none of these
55) The rank of a null matrix is
A) 0 B) 1 C) doesn't exist
D) none of these
56) If A=[aᵢⱼ]ₘₓₙ is a matrix and B is a matrix and B is a nonsingular square matrix of order r, then
A) rank of A is r
B) Rank of A is greater than r
C) rank of A is less than r
D) none of these
57) Which of the following is correct
A) determinant is a square matrix
B) determinant is a number associated to a matrix.
C) determinant is a number associated to a square Matrix.
D) none of these
58) let A be a square matrix of order n x n and K be a scalar. than |KA| equal to
A) K|A| B) |K| |A| C) Kⁿ |A| D) none
59) let A be a skew symmetric matrix of odd order, then |A| is equals to
A) 0 B) 1 C) -1 D) none
60) let A be a skew symmetric matrix of even order. than |A|
A) is a square B) is not a square
C) is always zero D) none
61) If A is an orthogonal matrix, then
A) |A|= 0 B) |A|=±1 C) |A|=±2 D) n
62) let A non-singular square Matrix. then |adj A| is equals to
A) |A|ⁿ B) |A|ⁿ⁻¹ C) |A|ⁿ⁻² D) N
63) Let A=[aᵢⱼ]ₘₓₙ be a square Matrix, and let cᵢⱼ be cofactor of aᵢⱼ in A. If C=[cᵢⱼ], then
A) |C|= |A| B) |C|= |A|ⁿ⁻¹
C) |C|= |A|ⁿ⁻² D) none
64) if A is a non-singular square matrix of order n, then the rank of A is
A) equals to n B) less than n
C) greater than n D) none
65) If A is a matrix such that there exists a square submatrix of order r which is non singular and every square submatrix of order r+1 or more is singular, Than.
A) rank (A)= r+1 B) rank (A)= r
C) rank (A) > r D) rank (A)≥ r+1
66) Let A be a matrix of rank r. then
A) Rank (A')= r B) Rank(A')<r
C)Rank (A')>r D) none of these
67) Let A=[aᵢⱼ]ₘₓₙ be a matrix such that aᵢⱼ = 1 for all I, j. then
A) rank(A)> 1 B) rank (A)=1
C) rank (A)= m D) rank(A)= n
68) If A is a non zero column matrix of order m x 1 and B is a nonzero row matrix of order 1xn then rank of AB equal to
A) m B) n C) 1 D) none
69) 1 2 3 0
the rank of matrix is 2 4 3 2
3 2 1 3
6 8 7 5
Is
A) 1 B) 2 C) 3 D) 4
70) if A is invertible Matrix, then det(A⁻¹) is equal to
A) det(A) B) 1/det(A) C) 1 D) n
71) if A= x y z and B= a h g
h b f
g f c
C= x
y
z
Find the order of ABC is..
A) 3x1 B) 1x1 C) 1x3 D) 3x3
72) If A and B are two matrices such that AB= B and BA= A, then, A² + B² =
A) 2AB B) 2BA C) A+B D) AB
73) IF A= 1 3
3 10 find Inverse of A
A) 10 3 B) 10 -3 C) 1 3 D) -1 -3
3 1 -3 1 3 10 -3 -10
74) If A= 0 1
1 0 then A⁴ is
A) 1 0 B) 1 1 C) 0 0 D) 0 1
0 1 0 0 1 1 1 0
75) If X= 3 -4
1 -1 then Xⁿ is
A) 3n -4n B)2+n 5-n
n - n n -n
C) 3ⁿ (4)ⁿ
1ⁿ (-1ⁿ) D) none
76) If A= 5 2
3 1 then inverse of A
A) 1 -2 B) -1 2
-3 5 3 -5
C) -1 -2 D) 1 2
-3 -5 3 5
77) For the Equations x+2y+3z=1, 2x+y+3z= 2, 5x+5y+9z= 4,
A) there is only one solution
B) there exist infinitely many solution
C) there is no solution D) none
78) If A= 3 1
-1 2 then A² is
A) 8 -5 B) 8 -5 C) 8 -5 D) 8 5
-5 3 5 3 -5 -3 -5 3
79) If A= 4 x+2
2x - 3 x+1 is symmetric, then x is equal to
A) 3 B) 5 C) 2 D) 4
80) If A+ B = 1 0
1 1 And
A - 2B = -1 1
0 -1 then metrix A is...
A) 1 1 B) 2/3 1/3 C) 1/3 1/3 D) n
2 1 1/3 2/3 2/3 1/3
81) Find the Inverse of -6 5
-7 6 is
A) -6 5 B) 6 -5 C) 6 5 D) 6 -5
-7 6 -7 6 7 6 7 -6
82) From the matrix equation AB= AC we can conclude B= C provided
A) A is singula
B) A is a nonsingular
C) A is symmetric
D) A is a square
83) If I₃ is the identity matrix of order 3, (I₃)⁻¹ is
A)0 B) 3 I₃ C) I₃ D) not necessarily exists
84) let a,b,c be positive real numbers, the following system of equation in x, y and z
x²/a² + y²/b² - z²/c² = 1, x²/a² - y²/b² + z²/c² = 1, -x²/a² + y²/b² + z²/c² = 1, has
A) no solution B) unique solution
C) infinitely many solutions
D) finitely many solution
85) if A and B are two matrices that A+B and AB are both defined then..
A) A and B are two matrices not necessarily of same order
B) And B are square matrices of same order.
C) number of columns of A= number of rows of B D) none
86) A and B are two square matrices of same order and A' denotes the transpose of A then
A) (AB)=B'A' B) (AB)'= A'B'
C) AB= 0 => |A|= 0 or |B|= 0
D) AB= 0 => A = 0 or B= 0
87) considered the system of equations a₁x + b₁y + c₁z= 0 , a₂x + b₂y + c₂z= 0, a₃x+ b₃y + c₃z = 0 if a₁ b₁ c₁
a₂ b₂ c₂ = 0
a₃ b₃ c₃ then the system has
A) more than 2 solution
B) 1 trivial and one non trivial solutions.
C) no solution
D) only trivial solution (0,0,0)
88) the system of linear equations x+y+z= 2, 2x+y-z= 3, 3x+ 2y+ kz= 4 has unique solution if
A) k≠ 0 B) -1<k<1
C) -2<k<2 D) k= 0
89) if A and B are square matrices of order 3 such that |A|= -1, |B|= 3, then |3AB| equals to
A) - 9 B) -81 C) -27 D) 81
EXERCISE - 2
Fill in the blanks to make the following statement Correct:
1) if (S)ᵢⱼ= k and A is a square Matrix of the same order, then AS= SA= ____
2) If A, B are square Matrices of the same order such that, ____, then (AB)ⁿ=AⁿBⁿ
3) the matrix 1 1 3
5 2 6
-2 -1 -3 is a nilpotent matrix of index___
4) The sum of two idempotent matrices A and B is idempotent if AB= BA= ____
5) If A= [aᵢⱼ] ₘₓₙ , B=[bᵢⱼ]ₙₓₚ be two matrices, then ((AB)')ᵢⱼ=____
6) If A and B are symmetric matrices, then A B is symmetric <=> ______
7) if A and B are symmetric matrices of order n, then AB+BA is ____
8) if A is a non-singular matrix, then Det (A⁻¹)= ____
9) if A is a skew symmetric matrix of odd order, then |A|=___
10) if A is an orthogonal matrix, then |A|= ____
11) if A and B are two invertible matrices such that AB= C, Det(B)= ____
12) if A is a square matrix of order n, then A(adj A) =(adj A)A= _____
13) if A= [aᵢⱼ], is a scalar matrix of order n such that aᵢⱼ = k, I= 1,2,...n, then |A|= ____
14) if A= diag (d₁ , d₂ , d₃...dₙ) is a diagonal Matrix, then |A|=___
15) if A and B be two non-null square Matrices such that AB is a null matrix, then A and B both are ___
16) If A is a nonsingular matrix, then adj (adj A)= ____
17) let AX = B be a system of n-equation with n-unknowns such that |A|= 0 and (adj A)B≠ 0, then the system is____
TRUE/FALSE
****************
1) A diagonal matrix is both an upper triangular and a lower triangular.
2) If S is a scalar matrix and A is a square matrix of the same order n, then AS= SA.
3) if A, B are two matrices such that AB and A+B are both defined, then A, B are square matrices of different orders.
4) If A,B are square matrices of the same order, then (AB)ⁿ=AⁿBⁿ
5) if A and B are m-rowed square matrices such that they commute, then
(A+B)ⁿ=ⁿC₀ Aⁿ + ⁿC₁Aⁿ⁻¹B+ ⁿC₂ Aⁿ⁻² B²+....ⁿCₙBⁿ.
6) If A, B are two Matrices such that AB= A and B A= B, then A and B are idempotent.
7) If B is an idempotent metrix and A= I - B, then AB= BA≠ 0
8) if A is a involutory matrix, then (I+A)(I-A)= 0
9) the matrix A = 1 1 3
5 2 6
-2 -1 -3
is a nilpotent matrix of index-4
10) If A, B are m-rowed square matrices, then AB=0 implies that at least one of A and B is the null matrix.
11) if AB= A and BA= B, then A, B are nilpotent.
12) Every Matrix can be written as the sum of a symmetric and a skew symmetric matrix.
13) All positive integral powers of a symmetric matrix are symmetric.
14) All positive integral powers of a skew symmetric matrix are skew-symmetric.
15) positive odd integral powers of a skew symmetric matrix are symmetric.
16) Positive even integral powers of a skew symmetric matrix are symmetric.
17) if A and B are two symmetric matrices of order n, then ABA is a symmetric matrix.
18) if A is a square Matrix of odd order, then |-A|= A.
19) if A is a nonsingular matrix, then Det (A⁻¹)= Det(A).
20) Every orthogonal matrix is invertible.
21) every invertible matrix is orthogonal.
22) Every skew-symmetric matrix of odd order is invertible.
23) if A and B are two invertible matrices, then (AB)⁻¹=A⁻¹B⁻¹
24) If A is non singular Matrix, then (A')⁻¹=(A⁻¹)'.
25) The inverse of a nonsingular diagonal matrix is a diagonal Matrix.
26) The product of two diagonal matrices of the same order is a diagonal Matrix.
27) If A is a non-singular symmetric matrix, then A⁻¹ is skew symmetric.
28) The adjoint of a diagonal matrix is a diagonal matrix .
29) A, B are n-rowed square Matrices such that AB= 0 and B is non singular. Then A≠ 0.
30) if A is an orthogonal Matrix, then A⁻¹ is also also orthogonal.
EXERCISE - 3
** Choose the correct alternative(s) in each of the following:
1) A) If A= 0 1 & B= 0 -i
1 0 i 0
then (A+B)² equals.
A) A²+B² B) A²+B²+ 2AB
C) A²+ B²+ AB - BA D) none
2) If A= 1. -2
5 3 then A+ A' equals
A) 2 3 B) 2 -4 C) 2 4 D) none
3 6 10 6 -10 6
3) if A=x 0 B) -2 1 C) 3 5 D) 2 4
1 y 3 4 6 3 2 1 and A+B= C - D, then
A) -3,-2 B)3,-2 C)3,2 D) -3,2
4) if A = i o and B= o -i
o i -i 0 then (A+B)(A-B) is equal
A) A²-B² B) A²+B²
C) A²- B²+ BA + AB D) none
5) If A', B' are transpose Matrices of the square matrices, then (AB)' is equal to
A) A'B' B) B'A' C) AB' D) BA'
6) If A= 1 -2 B= -1 4 C= 0 1
3 0 2 3 -1 0 then 5A - 3B + 2C=
A) 8 20 B) 8 -20 C) -8 20 D) 8 7
7 9 7 -9 -7 9 -20 -9
7) The adjoint of
1 1 1
1. 2 -3 is
2 -1 3
A) 3 -9 - 5 B) 3 -4 -5
-4 1 3 -9 1 4
- 5 4 1 -5 3 1
C) - 3 4 5
9 -1 -4 D) None
5 -3 -1
8) -5 2
If 1 -3 then adj A is equals to
A) - 3 -2 B) 3 -2
-1 -5 -1 5
C) 5 1 D) 3 2
2 3 1 5
9) If A= 0 1
0 0 I is the unit matrix of order 2 and a, b are arbitrary constants, then (aI + bA)² is equals to
A) a²I + ab A. B) a²I + 2ab A
C) a²I + b² A D) none
10) if the determinant of a square matrix of order n is expanded along a row (column), then it contains the sum of
A) n terms B) n² terms
C) n ! terms D) none
11) If w is one of the cube roots of unity, then 1 w w²
w 1 w² =
w² w 1
A) w B) w² C) 0 D) 1
12) If 0 < x < π/2 and
1 + sin²x cos²x 4 sin 4x
sin²x 1+ cos²x 4 sin4x sin²x cos²x 1+ 4sin 4x
then x is equals to
A) π/24, 5π/24. B) 7π/24, 11π/24
C) 7π/24, 11π/24 D) none
13) If A is 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)3 x 4 B) 3 x3 C) 4 x 4 D) 4 x 3
14) 1 1 0 1
If A= 0 1 B= 1 0
Then AB =
A) 0 0 B) 1 1. C) 1 0 D) -1 0
0 0 1 0 0 1 0 1
15) if the matrix AB is a null matrix, then
A) A= 0 or B= 0 B) A= 0 and B= 0
C) It is necessary that either A= 0 or B= 0
D) All the statement are wrong
16) If A= 2x 0 and A⁻¹= 1 0
x x -1 2 then x =
A) 1. B) 2 C) 1/2 D) none
17) if A= 5 x
y 0 and A= A' , then
A) x= 0, y= 5. B) x + y= 5
C) x= y. D) none
18) 1 0 0
If A= 0 1 0
a b -1 then A² is equal to
A) I B) 0 C) A D) - A
19) If X= 3 - 4
1 - 1 then Xⁿ =
A) 3n -4n B) 2+ n 5 - n
n - n n. - n
C) 3ⁿ (-4)ⁿ
1ⁿ (-1)ⁿ D) none
20) If A= i 0 B= 0. -1
0 - i 1 0 then which of the following is true.
A) AB= BA B) AB= 2BA
C) AB+ BA = 0 D) none
21) If the matrix 0 a 5
3 0 b
c 2 0 is skew-symmetric, then
A) a=3, b=2, c= 5 B) a=-3, b=-2, c=5
C) a=-3,b=-2,c=- 5 D) none
22) If A= i 0
0 i then A² =
A) 1 0 B) -1 0
0 -1 0 -1
C) 1 0 D) -1 0
0 1 0 1
23) If 2 -1
-1 2 and A²- 4A - nI = 0, then n is equal to
A) 3. B) - 3 c) 1/3 D) -1/3
24) If for a Square Matrix A= [aᵢⱼ] , aᵢⱼ = i² - j², then A is
A) unit matrix B) null matrix
C) Symmetric matrix
D) skew-symmetric matrix
25) If A= 0 1 and B= 1 0
-1 0 -1 0 then A² equals
A) I B) - I C) B D) - B
26) the matrix 1 2. 3
1. 2 3
-1 -2 -3 is
A) idempotent. B) nilpotent
C) involuntary. D) orthogonal
27) 1 0 1
If A= 0 0 1
a b 2 then aI +bA+2A²=
A) A B) -A C) abA D) none
28) 1 2. 2
If 3A = 2 1. -2
x 2 y is orthogonal, then x + y =
A) 3. B) 0. C) -3. D) 1
29) 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
30) Matrix theory was introduced by
A) Newton B) Cayley Hamilton
C) couchy. D) Euclid
31) If A= 1 - 1 B= a 1
2 -1 b -1
and (A+B)³= - (A²+ B²), then a, b is
A) 1, - 4 B) -1,4 C) -1, -4 D) 1, 4
32) If A =[a,b], B= [-b, -a] and C= [a
a] ,
then correct statement is
A) A= - B .B) A+ B = A- B
C) AC = BC D) CA = CB
33) If A= 2. 4. X= n B= 8
4 3 1 11 then AX = B, then n =
A) 1. B) 2 C) 4. D) none
EXERCISE---4
Metrix
1) 1 -2 4 0 -2 4
If A= 2 3 2 and B=1 3 2
3 1 5 -1 1 5
Then A+ B is
A) 1 - 2 4 B) 1 - 2 8
3 3 2 3 3 4
2 1 5 2 1 0
C) 1 - 4 8
3 6 4 D) none
2 2 10
2) If A² = 8A + kI where
A= 1 0
-1 7 then k is
A) 7 B) - 7 C) 1. D) -1
3) k 7 -2
The matrix 4 1 3
2 -1 2 is a singular matrix if k is
A) 2/5 B) 5/2 C) - 5 D) none
4) If the Matrix A= a b
c d then A² is..
A) a² b² B) a² +ab ab+ bd
c² d² ac+dc bc+d²
C) nonexistent. D) none
5) If A= k 0 and B= 1 0
1 1 5 1 such that A² = B then k is..
A) 1 B) -1. C) 4 D) none
6) If A= 2 -3 B= 1 5 m C= 2 4 1
1 k 0 2 3 1 -1 13 And AB= C then k, m is
A) 3, 4. B) 4,-3
C) no real values of k, m are possible D) none
7) If AB = 0 where
A= cos²x cosx sinx
cosx sinx sin²x and
B= cos²y cosy siny
cosy siny sin²y then |x-y| is equals to
A) 0 B) π/2 C) π/4. D) π
8) 0 - 4 1
If A= 2 k -3
1 2 -1 then inverse of A exist if
A) k≠ 4 B) k≠8 k= 4. D) none
9) The reciprocal matrix of
1 0 2
0 1 -1
1 2 1 is..
A) -3 - 4 2 B) 3 4 -2
- 1 1 -1 1 -1 1
1 2 -1 -1 -2 1
C) -3 - 1 1
-4 1 2
2 -1 -1 D) none
10) 1 -1 1
If A= 1 2 0
1 3 0 then the value of |adj A| is equals to
A) 5 B) 0 C) 1 D) None
11) sinx - cosx 0
If A= cosx sinx 0
0 0 1 then Inverse of A is equals to
A) A' B) A C) adj A D) none
12) 4 -1 -4
If A= 3 0 -4
3 -1 -3 then A² is equals to
A) A B) I C) A' D) none
13) cosx - sinx 0
If f(x)= sinx cosx 0
0 0 1 then f(x+y) is equals to
A) f(x) +f(y). B) f(x) - f(y)
C) f(x) . f(y). D) none
14) 1 w w²
If A= w w² 1
w² 1 w
And w w² 1
B= w² 1 w
w w² 1 And
C= 1
w
w² where w is the complex cube root of 1 then (A+B)C is equals to
A) 0 B) 1 0 0 C) 1 D) 1
0 0 1 0 0 1
0 0 0 1 1 1
15) 0 c -b a² ab ac
If A= -c 0 a & B= ba b² bc
b -a 0 ca cb c² then AB is equal to
A) 0 B) 1 C) 2I D) none
16) If A be matrix such that
B= 1 -2 & C= 6 0
1 4 0 6 And AB= C then A is
A) 2 4 B) -1 1 C) 4 2
1 -1 4 2 -1 1 D) N
17) the rank of the matrix
- 5 3 2
3 2 -5
4 -1 -3 is
A) 3 B) 2 C) 1 D) none
18) the rank of the matrix
1 2 3
K 2 4
2 -3 1 is 3 if
A) K≠18/11 B) K= 18/11
C) K= -18/11 D) none
19) The rank of the matrix
4 1 0 0
3 0 1 0
5 0 0 1 is
A) 4 B) 3 C) 2 D) none
20) If A= 1 x 1 B) 1 3 2
2 5 1
15 3 2
C) 1
2
x And ABC = 0 then x is
A) 2 B) -2 C) 14 D) none
22) x+y y & B= 2 C= 3
If A= 2x x-y -1 2 and AB= C then x.y is equals
A) -5 B) 5. C) 4 D) 6
23) 1 -2 3
2 -1 4
3 4 1 is a
A) rectangular Matrix
B) singular Matrix
C) square Matrix
D) Non singular matrix
24) 3 1 5 4 6
If A= -1 2 B= 4 1 2
0 6 -5 1 1
A) A+B exists B) AB exists
C) BA exists. D) none
25) 1 1 1
If A= 1 1 1
1 1 1 then
A) A³= 9A B) A³ = 27A
C) A+ A = A²
D) inverse of doesn't exist.
EXERCISE---5
1) If A= 2 1
5 3 then A² + 5I is
A)14 5 B) 19 5 C) 10 13 D) 39 13
25 19 25 14 13 39 13 10
2) If A= 3 2
5 0 then A² + 3A' is equals to
A) 28 12 B) 22 21 C) 28 21 D) N
30 10 30 0 21 10
3) If A= 2 3
1 2 then A⁻¹+ 2A' is...
A) 6 3 B) 6 -1 C) 2 3
1 6 5 6 9 1 D) N
4) If A= 3 5 and B= 2 1
1 2 3 -2 then (A⁻². B) is equal to
A)103 -36 B)103 37
37 -13 -36 -13
C) -57 59 D) -57 32
32 -33 59 -33
5) If A= 1 2
-1 -1 then A² - 4I is..
A) 0 B) I C) 5I D) -5I
6) 0 1 -1
If A= 4 -3 4
3 -3 4 then A²+ I is
A) 0 B) 2I C) -2I D) none
7) If A= 4 2
-1 1 then (A+ I)(A- 6I) is..
A) 0 B) 6I C) 12I D) - 12I
8) 4 0 0
If A= 0 4 0
0 0 4 then A³ - 16I is
A) -4I B) 48I C) -48I D) 0
9) If A= -i 0
0 -i then A²³ is ..
A) -A B) A C) I D) none
10) If A is a square matrix order 3 then det(A - A')= 0
B) A- A' is a symmetric matrix
C) A- A' is non singular Matrix D) n
11) Let A a square matrix of order 3 and A= B - C, where B is a symmetric matrix and C is a skew-symmetric matrix. If
A= 3 1 2
5 0 8
2 2 -1 then C is
A) 0 4 0 B) 0 2 0
-4 0 -6 -2 0 -3
0 6 0 0 3 0
C) 0 -4 0 D) 0 -2 0
4 0 6 2 0 3
0 -6 0 0 -3 0
12) If f(x) = x² + 4x -5 and A= 1 2
4 -3
Then f(A) is equals to
A) 1 0 B) -1 0 C) 8 4 D) -8 -4
0 1 0 -1 8 0 -8 0
13) Let A be a skew symmetric matrix of order 2, then
A) A⁻¹ does not exist
B) A⁻¹ exists and is a symmetric matrix
C) A⁻¹ exist and is non singular
D) none of these.
14) let A be a symmetric matrix of order 2 then
A) A⁻¹ exists and is a symmetric matrix
B) A⁻¹ exists and is diagonal Matrix
C) A⁻¹ existst but is not necessarily skew symmetric. D) none of these
15) If A and B are two square matrices such that AB= A and BA = B, then B² is equals to
A) I B) A C) B D) none
16) 0 c -b a² ab ac
If A= -c 0 a & B= ab b² bc
b -a 0 ac bc c²
Then AB is equals to
A) I B) -I C) 2I D) O
17) If A and B are two square matrices satisfying the conditions BA= A and AB = B then A² + B² =
A) AB B) 2AB C) BA C) A+B
18) B is a 2x2 matrix of all of whose elements are positive. If
B² = 17 8
8 17 then B is equal to
A) √17 2√2 B) 4 1 Or 1 4
2√2 √17 1 4 4 1
C) 1 4 Or 4 1
1 4 4 1 D) none
19) If the matrix A= a b
c d commutes with the metrix B= 1 0
0 0
then b² + c² is equal to
A) 0 B) 1 C) 2 D) none
20) If A be an orthogonal Matrix and P be a skew-symmetric matrix then A⁻¹PA is
A) an orthogonal matrix
B) symmetric matrix
C) skew symmetric matrix D) none
21) If A= 1 0 & B= 0 1
0 1 -1 0 then (aA+bB)(aA-bB) is equal to
A) a²A² - b²B² B) (a² - b²) A²
C) (a² + b²)A² D) (a² + b²)(A²-B²)
22) If A= 9 1 and B= 1 5
4 3 7 12 and 5A + 3B + 2C be a zero matrix then C is equals to
A) -24 -10 B)-24 - 10
-41/2 -51/2 -41 - 51
C) -24. -10 D) -24 -10
- 41 -51/2 -41/2 -51
23) If A= 1 -1 & B= a 1
2 -1 b -1 and if (A+B)²= A² + B³ then a² + b² is equals to
A) 5 B) 9 C) 17 D) none
24) let A= 1 0
-1 0 and (xA+ yI)²= A then x² + y² is equal to
A) 0 B) 1. C) 2. D) none
25) A and B are two square matrices of order 2 and
2A+ 3B= 8 3
7 6 A+ B'= 3 1
3 3
then B is equal to
A) 2 1 B) 1 0 C) 1 2. D) n
1 0 2 3 0 3
26) A and B are two square matrices of order 2 such that 2A+3B= I and A+ B= 2A', then A is
A) 5I B) -5I C) I/5 D) -I/5
27) If A= 3 1 -1
0 1 2 then A.A' is..
A) a symmetric matrix
B) a skew symmetric matrix
C) an orthogonal matrix D) none
28) If A is a 3x4 Matrix and B is such a matrix that both A'B and BA' are defined then the size of B is
A) 3x3 B) 4x4 C) 4x3 D) 3x4
29) which one of the following statement is true:
A) If A is symmetric then B'A B is also a symmetric
B) if A asymmetric the B'A B is skew symmetric
C) If A is skew symmetric then B'A B is symmetric
D) If A is skew symmetric then B'TA B is not skew symmetric.
30) Let A = 1 2 & B= 1 0
3 -5 0 2
If a matrix X satisfies the equation BX = A then the matrix X is equals
A)1/2. 2 4 B)1/2. 2 4 C)2 4 D) n
3 5 3 -5 3 -5
31) If A= i 0 & B= 0 -i
0 i -i 0 then (A+B)(A-B) is equal to
A) A²+B² B)A²-B² C) A²-2AB+B²D) N
32) If A= -1 -2 -2
2 1 -2
2 -2 1 then adj A=
A) 3A' B) A'/3 C) -3A'. D) -A'/3
33) If A= a b
c (bc+1)/2 and A+A⁻¹= KI, then K is equal to
A) (a²-bc+1)/a B) (a²+bc-1)/a
C) (a²-bc-1)/a D) (a²+bc+1)/a
34) If A= 3 2 B= -1 1 C= 2 -1
7 5 -2 1 0 4 and A.X. B = C then X is equals to
A) 16 3 B) 16 -3 C) -16 3 D) n
24 5 -24 5 24 -5
35) Let A = 1 tanx and B= 1 tanx
tanx 1 -tanx 1 then AB⁻¹is equal to
A) cos2x sin2x B) cos2x sin2x
sin2x cos 2x - sin2x cos2x
C) cos2x - sin2x D) cos2x - sin2x
-sin2x cos2x sin2x cos2x
36) If A and B are two symmetric matrices of order n, then (AB-BA) is
A) a symmetric matrix
B) a skew symmetric matrix
C) an identity Matrix D) none
37) Let A=[aᵢⱼ] 2x2 where aᵢⱼ= I +j then A² is equals to
A)13 8 B) 13 8 C) 13 18
8 13 8 25 18 25 D)n
38) Let A= -1 2
4 -3 then for any +ve integer n, Aⁿ is equals to
A) (-1ⁿ) 2n B) -n 2n
4n 3(-1)ⁿn -4n -3n
C) n -2 2+n
3+n n- 4 D) none
39) If A= 3 5 B= a b C) w 0
2 3 c d 0 w², and AB= Cⁿ, where n is an integer multiple of 3 and w is an imaginary cube root of unity, then a- b+ c - d
A) 3 B) -7 C) -3 D) 7
40) If A= 1 -1 1
-3 3 -3
-4 4 -4 then A² is
A) I B) A C) -A D) none
41) If adj 4 1
3 2 then A is ..
A) 2 -3 B)2 -1 C) 2 -3 D) none
-1 4 -3 4 -1 4
42) Let A(x)= cosx -sinx 0
sinx cosx 0
0 0. 1 then A(a+b) is equal to..
A) A(a)+ A(b). B)A+a)A(b)⁻¹
C) A(a). A(b) D) none
43) Let A= 0 0 -1
0 -1 0
-1 0 0 , then which of the following is the only correct statement ?
A) A²= I B) A is a zero matrix
C) A= - I D)A⁻¹is does not exist.
44) If, for the matrix A= 0 -4 1
2 x -3
1 2 -1
A) x≠4 B) x≠-4 C) x≠8 D) x≠-8
45) If A= 1/2. 0 1 B= 0 i
1 0 i 0 and C= 1 0
0 -1 and if X² = A² + B² + C² then X² is equal to
A) 9/4 B) -9/4 C) -3/4 D) 3/4
46) Let A be a 2x2 matrix aᵢⱼ=(3i-2i)/2, then Adj A is equal to
A) 1/2 1/2 B) 1/2 -1/2
-2 1 2 1
C) 1 1/2 D) 1 -1/2
-2 1/2 2 1/2
48) If B is a non-singular matrix and A is a square matrix then det(B⁻¹ AB) is equal to
A) det(A) B) -det(A)
C) det(B) D) det(B⁻¹)
49) If x, y are two +ve acute angles and if A(x)= cos²x cosx sinx
Cosx sinx sin²x satisfying A(x) A(y)O, then |x-y| is equal to
A) π B) π/2 C) π/4 D) 0 then
50) If A and B are 3x3 Matrices such that B'AB = A, then det B is equals to
A) 1 B)-1 C) ±1
D) ±1, provided A is non singular
51) Let A be a skew-symmetric and B a symmetric matrix of order k. If kAB + 5BA is skew-symmetric matrix and AB≠BA, then
A) k=5 B)-5 C) ±5 D) none
EXERCISE -- 6
1) If w is a complex cube root of unity, and A= w 0
0 w then A¹⁰⁰ is
A) A B) -A. C) O. D) none
2) 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. C) I + a²- bc= 0
3) The value of x for which the metric Matrix Product AB as
A= 2 0 7 & B= -x 14x 7x
0 1 0 0 1 0
1 -2 1 x -4x -2x equal to identity matrix is
A) 1/2. B) 1/3. C) 1/4 D) 1/5
4) If A³= O, then I+ A+ A² equal to
A) I - A B) B) (I-A)⁻¹C)(I+A)⁻¹D) N
5)Let A and B be 3x3 matrices. Then AB= 0 implies
A) A= O. B) |A|= 0 & |B|= 0
C) either |A|= 0 or |B|= 0
D) A= O & B= O
6) If A is a 3x3 skew-symmetric matrix, then |A| is given by
A) 0 B) -1. C) 1. D) none
7) If a,b and c are all different from zero such that 1/a+ 1/b + 1/c= 0, then A= 1+a 1 1
1 1+b 1
1 1 1+c is
A) symmetric B) non-singular
C) can be written as sum of symmetric and skew symmetric matrix D) none
8) if A is matrix such that A² +A+ 2I = O, then
A) A is non-singular
B) A is symmetric C) A≠ O
D) A⁻¹= - 1/2 (A+ I)
9) If AB= A and BA= B, then
A) A²= A. B) B²= B
C) A= I. D) B= I
10) If A is a non-singular matrix of order 3x3, then adj(adjA) is equal to
A) |A| A B) |A²| A
C) |A|)⁻¹A D) none
11) If A and B are two non-zero square matrices of the same order such that the product AB =O, then
A) both A and B must be singular
B) exactly one of them must be singular.
C) Atleast one of them must be non-singular. D) none
12) If A= 1 B= 1 -2 -1
2
-1
A) is not defined B) equal [-1]
C) equals to 1 D) is not invertible
4
1
13) If A and B are square matrices such that B= - A⁻¹BA, then (A+B)² is equal to
A) 0 B) A²+B²
C) A²+ 2AB+ B². D) A+ B
14) The number of values of k for which the system of equations (k+1)x + 8y = Kx + (k+3)y= 3k- 1
A) 0 B) 1 C) 2 D) none
15) if 2+x 3 4
1 -1 2
x 1 -5 is a singular matrix, then x is...
A) 13/25. B) -25/13
C) 5/13 D) -13/25
16) If the system of equations x+2y- 3z= 2, (k+3)z= 3, (2k+1)y + z= 2 is inconsistent, then value of k can be
A) - 3 B) -1/2 C) 1 D) 2
17) If the system of equations x- ky - z= 0, kx - y - z= 0, x+y - z= 0 has a non-zero solution, then value of k can be
A) -1 B) 1 C) 3 D) 5
18) The system of equation x +2y +3 z= 1, x - y +4z= 0, 2x+y +7z= 1
A) exactly one solution
B) only two solutions
C) no solution
D) infinitely many solutions
19) The value of a for which the system of equations x +y +z= 0, ax +(a+1) y +(a+2)z= 0, a²x+(a+1)³y +(a+2)³ z= 0 has a non-zero solution is...
A) 1 B) 0 C) -1 D) none
20) If A= a b c & B= q -b y
x y z -p a -x
p q r r -c z then
A) |A| =|B| B) |A| = - |B|
C) |A| = |B|
D) A is invertible if Ban is invertible.
21) If A is a square matrix, then A+ A' is a ____ matrix.
22) If A is a square matrix of order 3, then, |adj A|= ____
23) if Aₙ= cos n sin n
- sin n cos n , then Aₙ Aₚ = _____
24) If A and B are two invertible square matrices of the same order then adj(AB)=______
25) If A is a square matrix then adj(A') - (adj A') is a _____
26) If A and B are two symmetric matrices, then AB is symmetric matrix if and only if _____
27) The Matrix A= 1 1 3
5 2 6
-2 -1 -3 is a nilpotent matrix of index _____
28) If A and B are two matrices such that A+B and AB are both defined, then A and B must be___
29) If A is a square matrix and m is a positive integer such that Aᵐ= I, then A must be_____
30) If A and B are two idempotent matrices of the same order, such that A+ B is also an idempotent metrix than, AB+ BA is a____
31) A square matrix which is both symmetric and skew-symmetric must be____
32) If A is a 3x3 non-singular matrix such that A² = 2A, then |A|= ____
33) If A and B are two non-singular matrices of the same order such that Bʳ= I, for some positive integer r> 1. Then A⁻¹B ʳ⁻¹ A - A ⁻¹ B⁻¹A= ____
34) If abc≠ 0 then multiplicative inverse of A= a 0 0
0 b 0
0 0 c is ___
35) If A= a 0 0
0 a 0
0 0 a then |A| |adj A|= ____
36) If the system of equations x+ ay - z= 0, 2x - y +a z= 0, ax+y +2 z= 0 has a non-trivial solution, then value of a _____
37) let A= 2 3
-1 5 If A ⁻¹ = xA + yI , then x = ___ and y= ____
38) the system of equation Kx +y + z= 0, - x + ky +z= 0, -x- y +kz= 0 will have a non-zero solution if real values of k are given by ___
39) the matrix A= k -1 4
-3 0 1
-1 1 2 is invertible if and only if ____
40) If A and B are two matrices of the same order such that BA= I. Then inverse of A is____.
41) If A is skew symmetric matrix then A⁴ is a symmetric matrix T/F
42) If A is a diagonal Matrix, with non-zero elements along the main diagonal, then A is a non-singular. T/F
43) The adjoint of a diagonal matrix is a diagonal matrix. T/F
44) Let A= cos²x sinx
Cosx sinx sin²x and
B= cos²y cosy siny
cosy siny. sin²y
If the product AB is the null matrix, then x - y must be an integral multiple of π . T/F
45) If A is an orthogonal matrix, then A ⁻¹ = A'. T/F
46) If A= 1 0 B= 1 0
1/2 1 50 1 T/F
47) If A ⁻¹ =1 -1 2 then A=1 1/3 7
0 3 1 0 1/3 1
0 0 -1/3 0 0 3
T/F
48) If A(x,y)= cosx sinx 0
- sinx cosx 0
0 0 eʸ then A(x,y)⁻¹ = A(-x, -y). T/F
49) If A is a skew-symmetric matrix, then trace of A is zero. T/F
51) For each real x such that -1< x< 1, let A(x) denotes the matrix 1/√(1-x) . 1 - x
-x 1 show that A(x) A(y)= √(1+xy) A(z) where x,y belongs to R, -1<x, y< 1 and z= (x+y)/(1+xy)
52) Let A and B be two square Matrices, of the dimension, and let [A,B]= AB - BA. Show that for three 2x2 matrices A, B, C , [[AB], C]+ [[B,C], A] + [[C,A],B]= 0
53) If A= a₁₁ a₁₂ a₁₃
a₂₁ a₂₂ a₂₃
a₃₁ a₃₂ a₃₃ , we define trace of A to be the sum of its principal diagonal Matrix, that is, trace (A) = a₁₁+ a ₂₂ + a₃₃ if A and B are two 3x3 metrices, show that trace +AB)= trace (BA)
54) Prove that if A= 0 - tanx
tan x 0 (-π/2, < x < π/2), then U+A=
cos2x -sin2x
Sin2x cos2x (I+A)
55) Find all the matrices that commute with the matrix 1 2
3 4
56) let A= a b
c d where a, b, c, d belongs to R. Then show that A² = O if and only if A³= O.
57) If A and B are two invertible matrices of the same order such (AB)ⁿ = Aⁿ Bⁿ holds for three consecutive positive integers, Show that AB= BA.
58) Let A and B be two square matrices of the same order such that AB= BA, Aᵐ= 0 and Bⁿ= 0 for some positive integers m and n. show that there exists a positive integer r such that (A+B)ʳ= 0
59) using Matrix method find the value of u and v so that the system of equations. 2x -3y + 5z= 12, 3x + y + uz= v, x- 7y + 8z= 17 has
I) a unique solution
II) infinite solution
III) no solution
61) find all the values of C for which the equations 2x +3y= 3, (c+2) x + (c+4)y = c+6 , (c+2)²x + (c+4)²y = (c+6)² are consistent, also solve above equation for these values of of c.
62) Find inverse of
5 2 -6
7 -3 4
3 5 -12 by using elementary row operations.
EXERCISE --7
1) If A= 1 0 0
0 1 0
a b -1 then A² is equals
A) unit Matrix B) null matrix
C) A D) - A
2) If I = 1 0 & J = 0 1
0 1 -1 0 and
A= cosx - sinx
sinx cosx then A equals
A) cos xI + sin xI B) cos xI- sin xI
B) - cos xI+ sin xI D) none
3) A is a square Matrix then A - A' is...
A) a null matrix
B) a symmetric matrix
C) a skew-metric matrix
D) an invertible Matrix
4) A square Matrix which is both symmetric and skew symmetric is identity matrix is
A) the identity matrix
B) the null matrix
C) (A+A') + I Where A is any matrix. D) none
5) If A= 1 w w²
-w 1 w²
-w² w 1 where w is the complete cube root of unity, then A is
A) a symmetric matrix
B) a skew symmetric matrix
C) an invertible Matrix D) N
6) If A and B two symmetric matrices, then which one of the following is not necessarily true.
A) A+B is symmetric
B) A- B is symmetric
C) AB is a symmetric
D) BAB is symmetric
7) A and B are two square matrices of the same order and m is a positive integer, then
(A+B)ᵐ= m C ₀ Aᵐ + mC₁Aᵐ⁻¹B + mC₂Aᵐ⁻² B² +....mCₘ₋₁ ABᵐ⁻¹ + mCₘ Bᵐ if
A) AB= BA B) AB+ BA= 0
C) Aᵐ= O, Bᵐ= O D} none
8) If A= cosx - sinx 0
sinx cosx 0
0 0 0
Then A³ will be a null matrix if and only if
A) x= (2k+1)π/3, k belongs to I
B) x= (4k-1)π/3, k belongs to I
C) x= (3k-1)π/4, k belongs to I
D) none
9) If A= i -i &. B= 1 -1
-i i -1 1 then A⁸ equals to
A) 128B. B) 32B. C) 16B D) 8B
10) If x, y,z are three real numbers, and
1 cos(x-y) cos(a-z)
If A= cos(y-x) 1 Cos(y-z)
cos(z-a) cos(z-y) 1
Then
A) A is symmetry B)A is invertible
C) A is singular D). |A|
11) Let A and B be two invertible matrices. which one of the following are not true.
A) |A⁻¹|= |A|⁻¹. B) (AB)⁻¹= B⁻¹ A⁻¹
C) (A+B)⁻¹= A⁻¹+ B⁻¹
D) (3A)⁻¹= 3A⁻¹
12) if A is a square Matrix such that A³ - 3A²+ 5A +I= O where I denote the identity Matrix, then A⁻¹ is given by
A) 3A - A² - 5I. B) 3A + A² - 5I
C) A² - 3A + 5I. D) none
13) For a square Matrix, it is given that A'A= I, then A is
A) an orthogonal Matrix
B) a diagonal matrix
C) a symmetric matrix
D) invertible
14) let A and B two symmetric matrices. The product AB is symmetric if and only if
A) AB= BA. B) |AB|= 0
C)AB=O D) none
15) If A and B are two non-singular matrices such that AB= C, then |B| is equal to
A) |C|/|A|. B)|A|/|C| C) |C| D) none
16) A, B and C are three square matrices of the same order, then AB= AC => B = C if
A) A is singular
B) A is non-singular
C) A is symmetric
D) A is skew symmetric.
17) which one of the following is not correct ?
A) A² - B²= (A+B)(A-B)
B) (A')'= A
C) A² = I <=> (A+I)(A-I)= 0
D) A+ A' is symmetric
18) which one of the following statements are correct ?
A) A+ A' is symmetric for every square in matrix A.
B) A- A' is skew symmetric for every square Matrix A.
C) (AB)'= B'A' for all matrices A and B for which AB is defined.
D) AB= O => A= O or B= O.
19) If A= (aᵢⱼ)₃ₓ₃ is a skew symmetric matrix, then
A) (aᵢⱼ)= 0 or i.
B) A + A' is null matrix
C) |A|= 0 D) none
20) If A and B are 3x3 metrices and |A|= 0, then
A) |AB|= 0 => |B|= 0
B) AB= O => B= O
C) |A⁻¹|=|A|⁻¹. D) |A+ A|= 2|A|
** FILL THE BLANKS:
21) If A is an upper triangular Matrix then A⁻¹ is_____
22) if A= (aᵢⱼ)₃ₓ₃ satisfy the relation A⁻¹ = A²+ 5I, then A³ +5A= ___
23) If A = sinx i cosx
i cosx sinx then all x belongs to R, A⁻¹____
4) If A and B are two square matrices of same order then (A+B)(A-B)= _____
5) Let Cₖ = ⁿCₖ for 0 ≤ k ≤ n, and Aₖ = Cₖ₋₁ 0
0 Cₖ for k≥ 1. If A₁ A₂ + A₂ A₃ +......+ Aₙ₋₁ Aₙ = k₁ 0
0 k₂ then k₁= ____ and k₂= ____.
6) If A is a square matrix such that I + A + A²= 0, then A⁻¹=___
7) Let a and b be two real numbers such that a> 1, b> 1. If A= a 0
0 b , then, lim ₓ→∞ A⁻ⁿ=__
8) If A= i 0
0 i then inverse of A is __
9) If A= a b
c d satisfies the equation x² -(a+d)x + k= 0, then k__
10) If A⁵= O and Aⁿ≠ I for 1≤n≤4, then (I - A)⁻¹=_____
11) If D₁ and D₂ are two 3x3 diagonal matrices, then D₁² +D₂² is ___
12) The number of distinct matrices that can be formed by using 10 real numbers is____.
13) If A is a square matrix such that A² = A, then (I+A)³ - 7A=___
14) If A is square matrix such that A² = A, then (I+A)³ - 7A= ___
15) If A is a nonsingular matrix such that AA' = A'A, and B= A⁻¹, then BB'= ____.
16) Let B and C be two square Matrices such that BC = CB and C² = 0. If A= B + C, then A³ - B³ - 3B²C= ____.
17) IF A is a non-singular matrix and B= A⁻¹CA, then |B|=___.
18) If a+b+c= 0, then the system of equations ax+by+cz=0; bx+cy +az =0; cx+ay+bz=0 has ___
19) The system of equations kx+y+z=1; x+ky +z =1; x+y+kz=12 has a unique solution if k is different from___.
20) If 1 3 k+2
2 4 8
3 5 10 is a singular then k= ____
** TRUE/FALSE
1) A square Matrix can always be written as sum of a symmetric and a skew symmetric matrix.
2) If D₁ and D₂ are two diagonal matrices of order 3x3, then D₁D₂ is a diagonal matrix.
3) If x belongs to R then
A= Cosx sinx B= Sinx - cosx
- sinx cosx cosx sinx
And the relation cosx . A+ sinx . B is an identity Matrix.
4) The Equation AX= B where
A= 4 6 & B= 1 1
6 9 1 1 has infinite number of solutions.
5) The determinant of an orthogonal matrix is equal to ± 1
6) if A is a square Matrix such that A² = I and B= (I+A)/2, then B² = B.
7) If A is its own inverse and B= (I+A)/2, then B is its own inverse.
8) If A is a Square Matrix such that A⁻¹ does not exist, then there is no positive integers m such that inverse of Aᵐ exists.
9) Every invertible matrix is orthogonal.
10) A= cosx - sinx
sinx cosx is an orthogonal Matrix.
*** Prove that::
1) Let A.B be two square matrices of same dimensions, and Let A* B = (AB+AB)/2.
2) Prove that
A) A * B = B* A
B) A* A = A²
C) A * I= A
D) A* (B+C)= A * B + A* C
E) A * (B+C)= 1/4(ABC + ACB + BCA + CBA)
F) (kA)* B= A *(kB)= k(A* B) where k is any real number.
2) If a, b, c are distinct and x,y,z are distinct, show that the product MN where
M= ax a a 1 & N= 1 1 1
by b y 1 -x -y -z
cz c z 1 ax by cz is non-singular.
3) Prove that the matrix A= a b
c d satisfies the equation x² - (a+d)x + ad - bc = 0.
4) Show that there cannot exist two 3x3 square matrices A and B such that AB- BA = I, where I denotes identity matrix.
5) If A and B are two non-singular Matrices of the same order. Show that the following statements are equivalent:
A) AB= BA
B) A⁻¹B=B A⁻¹
C) AB⁻¹= B⁻¹A
D)A⁻¹B⁻¹= B⁻¹A⁻¹
6) Let A be a no-singular matrix of order 3. Find all complex numbers c such that det(cA)= det(A).
7) Find all the matrices that commute with the matrix 7 - 3
5 - 2
8) If A, B, A+ I and A+B are Idempotent matrices, Show that AB= BA.
9) For what value of K do the following system of equation possess a for non-trivial solution (i.e., not all zero) solution over the set of rational number x+ ky+ 3z= 0, 3x+ ky - 2z= 0, 2x+ 3y- 4z= 0. For that value of k, find all solutions of system.
10) If a, b, c are distinct, show that the matrix a b c
b c a
c a b is invertible.
11) show that the system of equations 3x-y+4z= 3, x+ 2y- 3z=-2; 6x + 5y+ Kz= -3 has at least one solution for any real k, find the set of solutions if k=-5,
12) For what real values of K, the system of equations x+ 2y+z= 1, x+ 3y + 4z= k; x + 5y+ 10z= k² has solution ? find solution in each case.
13) Using Matrix method find the values of K and M so that the system of equations 2x- 3y+5z= 12, 3x+ y +kz= M; x -7y+ 8z= 17 has
A) unique solution
B) infinite solution
C) no solution.
14) Find the real values of R for which the following system of linear equations has a non trivial solution. Also, find the non trivial solutions: 2rx- 2y+3z= 0, x+ ry + 2z=-0 ; 2x + rz= 0.
15) Prove that
A) the product of two orthogonal matrices is an orthogonal matrix,
B) The inverse of an orthogonal matrix is orthogonal.
16) A square matrix A is said to be involuntary A² = I.
If a square matrix P is such that P²= P, then show that A= 2P - I is involuntary and B= 1/2 (A+ I) satisfies the condition B²= B.
17) Show that a matrix A= (aᵢⱼ)₃ₓ₂ which commute with all the 3x3 diagonal matrices is itself a diagonal matrix.
18) If A= 1 0
1 2 show that A²= 3A - 2I. Using this result show that A⁸= 255A - 254I.
19) If A= p q
0 1 show that
A⁸= p⁸ (p⁸ -1)q/(p-1)
0 1
20) If A= 1 a
0 1 show that
A⁴⁰= 1 40a
0 1
21) Use elementary row operations to commute the Inverse of
A) 1 2 -1 B) 2 0 0
0 2 1 8 -1 0
0 0 5 4 0 2
Also verify your answer.
Continue.......
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