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 –

(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.

(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.

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.