Properties of Multiplication of Matrices

Here you will learn properties of multiplication of matrices, positive integral powers of square matrix and matrix polynomial.

Let’s begin –

Also Read : Multiplication of Matrices – Examples & Definition

Properties of Multiplication of Matrices

(a) Matrix multiplication is not commutative in general i.e AB $\ne$ BA.

(b) Matrix multiplication is associative i.e. (AB) C = A (BC), whenever both sides are defined.

(c) Matrix multiplication is distributive over matrix addition i.e

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

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

(d) If A is an $m\times n$ matrix, then $I_m$ A = A = A $I_n$.

(e) If A is $m\times n$ matrix and O is a null matrix, then

(i) $A_{m\times n}$ $O_{n\times p}$ 

(ii) $O_{p\times m}$ $A_{m\times n}$

i.e. the product of the matrix with a null matrix is always a null matrix.

Positive Integral Powers of a Square Matrix

for any square matrix, we define

(i) $A^1$ = A

(ii) $A^{n+1}$ = $A^n$.A, where n $\in$ N

It is evident from this definition that $A^2$ = AA, $A^3$ = $A^2$A = (AA) A. etc.

It can be easily shown that

(i) $A^{m}$$A^{n}$ = $A^{m+n}$ and,

(ii) $(A^{m})^n$ = $A^{mn}$ for all m, n $\in$ N.

Matrix Polynomial

Let f(x) = $a_0x^n$ + $a_1x^{n-1}$ + $a_2x^{n-2}$ + ….. + $a_{n-1}x$ + $a_n$ be a polynomial and let A be a square matrix of order n. Then,

f(A) = $a_0A^n$ + $a_1A^{n-1}$ + $a_2A^{n-2}$ + ….. + $a_{n-1}A$ + $a_nI_n$ 

is called a matrix polynomial.

for example, if f(x) = $x^2$ – 3x + 2 is a polynomial and A is a square matrix, then f(A) = $A^2$ – 3A + 2I is a matrix polynomial.