Show that AB = BA in each of the following cases: A = [(1,2,1)(3,4,2)(1,3,2)] B = [(10,-4,-1)(-11,5,0)(9,-5,1)]

Question

Show that AB = BA in each of the following cases:

A = \(\begin{bmatrix} 1 & 2 & 1 \\[0.3em] 3 & 4 & 2 \\[0.3em] 1 &3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 10 & -4 & -1 \\[0.3em] -11 & 5 & 0 \\[0.3em] 9 &-5 & 1 \end{bmatrix}\)  

A = [(1,2,1)(3,4,2)(1,3,2)]

B = [(10,-4,-1)(-11,5,0)(9,-5,1)]

Answer 1

Given : A = \(\begin{bmatrix} 1 & 2 & 1 \\[0.3em] 3 & 4 & 2 \\[0.3em] 1 &3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 10 & -4 & -1 \\[0.3em] -11 & 5 & 0 \\[0.3em] 9 &-5 & 1 \end{bmatrix}\)

Matrix A is of order 3 x 3 and Matrix B is of order 3 x 3

To show : matrix AB \(\neq\) BA

Formula used :

If A is a matrix of order a x b and B is a matrix of order c x d ,

then matrix AB exists and is of order a x d ,if and only if b = c

If A is a matrix of order a x b and B is a matrix of order c x d ,

then matrix BA exists and is of order c x b , if and only if d = a

For matrix AB, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 x 3

For matrix BA, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 x 3

Matrix BA =

Matrix AB \(\neq\) BA