INVERSE MATRICES

Evaluation adjoint and inverse Matrix using determinants and elementary transformations

MINORS AND COFACTORS

MINOR: Let A = [aij] be a square matrix of order n. Then the minor Mij of aij in A is the determinant of the square sub-matrix of order (n-1) obtained by leaving ith row and jth column of A

1. If , we have

M11 = Minor of A11 = 2, M12 = Minor of A12 = —3,

M21 = Minor of A21 = —7, M22 = Minor of A22 = 4

COFACTORS: Let A = [aij] be a square matrix of order n. Then the cofactor Cij of aij in A is equal to (—1) i+j. times the determinant of the sub-matrix of order (n-1) obtained by leaving ith row and jth column of A.

If follows from this definition that

Cij =Cofactor of aij in A

= (-1) i+j Mij, where Mij is minor of aij in A

Thus, Cij = Mij if i+j is even

And, Cij = - Mij if i+j is odd.

2. If , then we have

C11 = (-1)1+1 M11= M11 = = 2,

C12 = (-) 1+2M12 = - M12 = = 7

C13 = (-1)1+3 M13 = M13 = = 8,

C23 = (-1)2+3 M23 = - M23 = = 8 etc.

ADJOINT OF A SQUARE MATRIX

DEFINITION

Let A = [aij] be a square matrix of order n and let Cij be cofactor of aij in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A.

Thus, adj A = [Cij] T.

(adj A) ij = Cij = Cofactor of aji in A.

Find the adjoint of matrix.

SOLUTION

We have,

Cofactor of A11 = s,

Cofactor of A12 = —r,

Cofactor of A21 = -q

And, Cofactor of A22 = p.

It is evident from this example that the adjoint of a square matrix of order 2 can be easily obtained by interchanging the diagonal elements and changing signs of off-diagonal elements.

If , then by the above rule, we have,

Find the adjoint of matrix

SOLUTION

Let Cij be cofactor of aji in A. Then cofactors of elements of A are given by

PROPERTIES OF ADJOINT The following are some properties of adjoint of a square matrix which are states as theorems.

Let A be a square matrix of order n. Then

A (adj A) = |A|In = (adj A) A

Let A be a non-singular square matrix of order n. Then

|adj A| = |A|n-1

If A and B are non-singular square matrices of the same order, then

adj AB = (adj B) (adj A)

If A is an invertible square matrix, then

adj AT = (adj A) T.

If A is a non-singular square matrix, then

adj ( adj A) = |A|n-2 A

If A is a symmetric matrix, then adj A is also a symmetric matrix.

INVERSE OF A MATRIX

DEFINITION A square matrix of order n is invertible if there exists a square matrix B of the same order such that

AB = I„ = BA

In such a case, we say that the inverse of A is B and we write,

A-1= B

PROPERTIES OF INVERSE

The inverse of a matrix has the following properties

Every invertible matrix possesses a unique inverse.

A square matrix is invertible iff it is non-singular.

The inverse of A is given by

1. AB=AC => B=C
2. BA=CA => B=C

Clearly, AB = BC but B ≠ C.

(Reversal law) if A and B are invertible matrices of the same order, then AB is invertible and

(AB)-1 = B-1 A-1

If A is an invertible square matrix, then

AT is also invertible and (AT)-1 = (A-1) T.

If the non-singular matrix A is symmetric, then A-1 is also symmetric.

If A is a non-singular matrix, then