Saturday, 14 June 2025

DEFINITION XII+

MATRICES 

DEFINITION:
A Set of mn numbers (real or imaginary) arranged in the form of a rectangular array of m rows and n columns is call ped Matrix.

ORDER OF A MATRIX 
A matrix having m rows and n columns is called a of order mxn.
a₁₁   a₁₂  a₁₃ ....a₁ⱼ ....a₁ₙ
a₂₁   a₂₂  a₂₃.....a₂ⱼ.....a₂ₙ
 |       |      |         |         |
aᵢ₁    aᵢ₂  aᵢ₃......aᵢⱼ......aᵢₙ
aₘ₁  aₘ₂ aₘ₃ ....aₘⱼ.....aₘₙ]ₘₓₙ
Or A= [aᵢⱼ]ₘₓₙ, 1≤ i≤ m, 1 ≤ j≤ n, i,j ∈ N

TYPES OF MATRIX 
1. Rectangular matrix : If the number of rows in the columns of a matrix are not equal, then the matrix is called rectangular Matrix, i.e., m≠ n

2. Row Matrix : A matrix containing only one row is called a row matrix. i.e., m= 1

3. Column Matrix: A matrix containing only one column is called a column matrix i.e., n= 1

4. Null or Zero Matrix : If every element of a matrix is zero, then it is called a zero or null matrix.

5. Square matrix: A matrix in which the number of rows is equal to the number of columns is called square Matrix p.

6. Diagonal Matrix: A square Matrix in which every non diagonal element is zero is called a diagonal Matrix.

7. Scalar matrix : A square Matrix in which all the elements in the diagonals are equal and rest of the elements are zero is called a scalar matrix.

8. Identity or unit matrix: A square in which each diagonal elements is unity and all other elements are zeros is called a unit matrix. It is denoted by I.

9. Upper triangular matrix : A square Matrix in which all the elements below the diagonal are zeros is called an upper triangular matrix.

10. Lower triangular matrix: A square Matrix in which all the elements above the diagonal are zeros is called a lower triangular matrix.

11. Equal matrix : Two matrices are said to be equal if they are of the same order and each element of one matrix is equal to the corresponding element of other matrix.

OPERATION ON MATRICES 

• Matrix addition : Let A= [aᵢⱼ] and B=[bᵢⱼ] be two mxn matrices. Then the sum A+ B of A and B is defined as A+ B= [cᵢⱼ]ₘₓₙ , where cᵢⱼ = aᵢⱼ + bᵢⱼ for each i and j.

• Substraction of a matrix : Let A= [aᵢⱼ]ₘₓₙ and B=[bᵢⱼ]ₘₓₙ be two matrices of same order, then the difference A - B of A and B is defined as A - B = A +(- B)= [aᵢⱼ - bᵢⱼ]ₘₓₙ for each i and j.

• Negative of Matrix : Let A= [aᵢⱼ]ₘₓₙ be a matrix, then the negative of the matrix A is defined as the matrix [-aᵢⱼ]ₘₓₙ and is denoted by - A. Thus negative of a matrix is obtained by making all the elements of the given matrix negative .

• Scalar multiplication of a matrix: Let [aᵢⱼ]ₘₓₙ be a matrix and k is a scalar, then the Matrix obtained by multiplying the individual element of the matrix A by k is called the scalar multiplication of A by k and is denoted by kA or Ak.

• Multiplication of two matrices : Let A= [aᵢⱼ]ₘₓₙ and B=[bᵢⱼ]ₙₓₚ be two matrices, then the product AB= C = [cᵢₖ]ₘₓₚ where cᵢₖ = aᵢ₁ b₁ₖ + aᵢ₂ b₂ₖ + ....+ aᵢₙ bₙₖ.


Properties of matrix Addition and Multiplication 
• Commutative law
If A and B are two matrices, then 
A+ B= B+ A and AB ≠ BA 

Associative law
If A, B and C are three matrices, then 
(A+ B)+ C = A+(B + C) and (AB)C = A(BC(

• Distributive law
A(B+ C)= AB+ AC and (A+ B)C = AC+ BC 

• Existence of Identity
 The null matrix is the identity element for Matrix edition i.e., A+ O = A= O+ A
If A is a square Matrix, then IA = A = AI, where I is identity matrix of same order.

• Existence of inverse :
For every matrix A= [aᵢⱼ]ₘₓₙ there exists a matrix [-aᵢⱼ] ₘₓₙ denoted by - A, such that A+ (-A)= O= (-A)+ A

• If A is mxn matrix and O is a null Matrix, then
i) Aₘₓₙ Oₙₓₚ = Oₘₓₚ
ii) Oₚₓₘ Aₘₓₙ = Oₚₓₙ 

TRANSPOSE OF A MATRIX 
• Let A= [aᵢⱼ]ₘₓₙ matrix. Then the transpose of A, denoted by Aᵀ or A', is an nxm matrix such that Aᵀ=  [aᵢⱼ]ₘₓₙ and i= 1,2,....m and j= 1,2,....,n.
Thus Aᵀ is obtained from A by changing its rows in to columns and its columns into rows.

Properties of Transpose 
• Let A and B be two matrices, then 
i) (Aᵀ)ᵀ= A
ii) (A+ B)ᵀ= Aᵀ + Bᵀ,where A and B being of the same order.
iii) (kAᵀ)= kAᵀ, k  be any scalar (real or Complex)
iv) (AB)ᵀ = BAᵀ , A and B being conformable for product AB . (Reversal law)
v) (ABC)ᵀ= CᵀBAᵀ.


ELEMENTARY OPERATION 
There are six operations (transformations) on a matrix , three of which are due to rows and three due to columns, which are known as alimentary operations or transformations .
i) The multiplication of the elements of any row or column by a non zero number. Rᵢ --> R
The corresponding column operation is denoted by Cᵢ --> kCᵢ where k≠ 0
For example, applying R₂ --> (1/4) R₂ to A
A= a₁₁   a₁₂  we get a₁₁      a₁₂
      a₂₁   a₂₂              a₂₁/4  a₂₂/4

ii) The interchange of any two rows or two columns i.e., Rᵢ <---> Rⱼ and Cᵢ <--> Cⱼ.
For example, applying R₁ <--> R₃ to A
     a₁₁   a₂₁   a₁₃               a₃₁   a₃₂   a₃₃
A=a₂₁   a₂₂    a₂₃ we get  a₂₁   a₂₂   a₂₃
     a₃₁   a₃₂    a₃₃               a₁₁   a₁₂   a₁₃

iii) The addition to the elements of any two rows or column, the corresponding elements of any other row or column multiplied by any non-zero number 
i.e., Rᵢ---> Rᵢ + kRⱼ or Cᵢ---> Cᵢ + kCⱼ.
For example applying C₁--> R₁ - kR₂ to A

A= a₁₁   a₁₂  we get a₁₁ - ka₂₁  a₁₂ - ka₂₂
      a₂₁   a₂₂              a₂₁            a₂₂

 
SOME SPECIAL MATRICES 
symmetric matrix : A square Matrix A= [aᵢⱼ] is called a symmetric matrix if  aᵢⱼ = aᵢ for all i, j.

Skew-symmetric matrix: A square matrix A= [aᵢⱼ] is a Skew-symmetric matrix if  aᵢⱼ = - aⱼᵢ for all i, j.

Orthogonal matrix : A square matrix A is called an orthogonal matrix if AAᵀ =AᵀA = I.

Equivalent matrices: Two matrices A and B are equivalent if one can be obtained from the other by a sequence of elementary row transformations .


INVERTIBLE MATRICES 
if A is a square matrix of order n and if there exists another square Matrix B of the same order n, such that AB = BA= I, then B is called the inverse matrix of A and it is denoted by A⁻¹.  In that case A is said to be invertible.

Uniqueness of inverse 
Inverse of a square matrix, if it exists , is unique .
Let A=[aᵢⱼ] be a square matrix of order n. If possible, let B and C be two inverses of A.
To show: B= C
Since B is the inverse of A
=> AB= BA = I
Since C is also the inverse of A
=> AC= CA= I
Thus B= BI = B(AC)= (BA)C = IC = C
If A and B are invertible matrices of the same order, then (AB)⁻¹= B⁻¹A⁻¹.

Inverse of a Matrix by Elementary Operation 
To find A⁻¹.using elementary row operations , write A= IA and apply a sequence of row operations on A= IA till we get, I= BA. The matrix B will be the inverse of A. Similarly, if we wish to to find A⁻¹ using column operations , then write A= AI and apply a sequence of colum operations on A= AI till we get, I= AB 




DETERMINANTS 

DEFINITION: Corresponding to each square Matrix A= (aᵢⱼ)ₘₓₙ there is an associated expression , called the determinant of A, denoted by det A or |A|, written as 
|a₁₁   a₁₂  a₁₃ ....a₁ₙ
a₂₁   a₂₂  a₂₃.....a₂ₙ
....    ....   ... ...   ...

aₙ₁  a₂   a₃ ....aₙ|ₘₓₙ

Value of determinant of order 1
Let A= [a] be Matrix of order 1, then determinant A is defined to be equal to a.

•  Value of determinant of order 2

|a₁₁   a₁₂ 
 a₂₁    a₂₂| = (a₁₁ a₂₂ - a₂₁ a₁₂)
 

Value of a determinant of order 3 or more 
The value of a determinant is the sum of the product of a row (or a column) with their corresponding factors.
So, det A= ⁿᵢ₌₁∑ aᵢⱼ Cᵢⱼ 
 
Properties of determinant
i) det Iₙ= 1, where Iₙ is unit/identity where unit/identity matrix of order n.

ii) det 0ₙ= 0, 0 is null Matrix (square matrix of any order).

iii) A= (aᵢⱼ)ₙₓₙ  then |A|= |A'| (Reflection Property).

iv) det (AB )= det A . det B, where A & B are matrices of same order.

v) det(kA)= kⁿ det A, if A is of order n x n.

vi) det (Aⁿ)= (det A)any if n ∈ I⁺

vii) |A|= 0 iff
      a) Any two rows or columns are identical.
      b) Any two rows or columns are in proportion.
      c) Each elements of any row/ column is zero.

viii) if each element of a row/column of a determinant is multiplied by k then value of new determinant is k times the original Determinants.

ix) Determinant of a diagonal matrix is the product of its diagonal elements.

x) If two row/columns of a determinant are interchanged, then the determinant retains its absolute value but its sign is changed.

xi) if each element of a row/columns of a determinant is expressed as a sum of two or more terms, then determinant can be expressed as the sum of two or more determinants.

xii) if any row/column of a determinant, a multiple of another row/column is added, then the value of determinant does not change.

xiii) The sum of product of the elements of any row/column of a determinant with cofactors of the corresponding elements of any other row/ column is zero.


APPLICATION OF DETERMINANTS 

• Area the triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is given by 
(1/2) | x₁      y₁     1
            x₂      y₂     1
            x₃      y₃      1

• If area is zero then there points are collinear.


MINORS AND COFACTORS 

Minor of  aᵢⱼ in |A|: The minor of an element aᵢⱼ in |A| is defined as the value of the determinant obtained by deleting the iᵗʰ row and jᵗʰ column of |A|, and it is denoted by Mᵢⱼ .

Cofactor of aᵢⱼ in |A|: The cofactor Cᵢⱼ of an element aᵢⱼ is defined as Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ.

ADJOINT OF A MATRIX 

• Adjoint of a matrix A= (aᵢⱼ)ₓₙ is defined as adj A= [Cᵢⱼ]ₓₙ, where 
Cᵢⱼ represents cofactor of aᵢⱼ in |A|.

Properties of Adjoint of a matrix

• For every square matrix
A(adj A)= (adj A)= |A|. I.

• If |A|=0 matrix A is called singular else non singular.

•  if A and B are non singular matrices of the same order, then AB and BA are also non singular matrices of same order.

• (adj AB)= (adj B).(adj A)

• (adj A)' = adj A'.

• let A be n x n matrix, then 
i) |adj A|= |A|ⁿ⁻¹
ii) adj (adj A)= |A|ⁿ⁻²A
iii) |adj (adj A|= |A|(ⁿ⁻¹)^2

INVERSE OF A MATRIX 

A non zero matrix A of order n is said to be invertible if there exists a square matrix B of order n such AB = BA= I. We say A⁻¹= B.

• A matrix A is invertible if |A|≠ 0.

• A⁻¹= 1/|A|.   (adj A).

• If AB= AC => B = C if |A|≠ 0.

• (AB)⁻¹= B⁻¹A⁻⅓.

• (Aᵀ)⁻¹= (A⁻¹)ᵀ.

• If A is an invertible symmetric matrix then A⁻¹ is also symmetric.

SOLUTION OF SYSTEM OF EQUATIONS

let AX = B be the given system of equations:
    i) If |A|≠ 0, the system is consistent and has one unique solution.
   ii) If |A|= 0 and (adj A)B ≠ 0, then the system is inconsistent so it has no solution.
   iii) If |A|= 0 and (adj A)B=0, then the system is consistent but has infinitely many solutions. 














































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