# 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 –

## 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.