MULTIPLICATION OF MATRICES

Definition

Two matrices A and B are conformable for the product AB if the number of columns in A (pre -multiplier ) is same as the number of rows in B (post-multiplier ).Thus , if A =[a_ij]_m×n and B =[b_ij]_n×p are two matrices of order m×n and n×p respectively , then their product AB is of order m×p and is defined as

PROPERTIES OF MATRIX MULTIPLICATION

(1)Matrix multiplication is not commutative in general .

(2)Matrix multiplication is associative ,that is , (AB)C =A(BC) , whenever both sides are defined .

(3)Matrix multiplication is distributive over matrix addition ,that is ,

(i)A(B +C)=AB +AC

(ii)(A+ B)C =AC +BC   whenever both sides of equality are defined .

(4)If A is an m×n matrix , then l_mA=A=Al_n

(5)The product of two matrices can be the null matrix while neither of them is the null matrix .

(6)If A is m×n matrix and O is a null matrix , then

(i)Am×n On×p = Om×p

(ii)Op×m Am×n = Op×n

That is , the product of the matrix with a null matrix is always a null matrix .

(7)In the case of matrix multiplication if AB=O , then it does not necessarily imply that BA =O .

Positive Integral Powers of a Square Matrix

For any square matrix , we define

(i)A¹ =A

(ii)A⁽ⁿ⁺¹⁾= Aⁿ .A  where n is any natural number .

(iii)A^m .Aⁿ = A^(m+n)

(iv)(A^m)ⁿ = A^mn

Matrix Polynomial

Let f(x)= a₀xⁿ +a₁x^n-1 +a₂x^n-2 +....+a_n-1x +a_n be a polynomial and let A be  a square matrix of order n .Then ,

     f(A) = a₀Aⁿ + a₁A^n-1 + a₂A^n-2 +....+a_n-1A +a_nl_n

Is called a matrix polynomial .

For example , if f(x) = x² -3x +2 is a polynomial and A is a square matrix , then

A² -3A +2I is a matrix polynomial.