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Thursday, July 28, 2016

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.
  1. 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

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