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

A = \(\begin{bmatrix} 1 & 3 & -1 \\[0.3em] 2& 2 &-1 \\[0.3em] 3 &0 & -1 \end{bmatrix}\) and B = \(\begin{bmatrix} -2 & 3 & -1 \\[0.3em] -1& 2 &-1 \\[0.3em] -6 &9 & -4 \end{bmatrix}\)

A = [(1,3,-1)(2,2,-1)(3,0,-1)]

B = [(-2,3,-1)(-1,2,-1)(-6,9,-4)] 

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Given : A = \(\begin{bmatrix} 1& 3 &-1 \\[0.3em] 2& 2 & -1\\[0.3em] 3 & 0 & -1 \end{bmatrix}\) and B = \(\begin{bmatrix} -2& 3 &-1 \\[0.3em] -1& 2 & -1\\[0.3em] -6 & 9& -4 \end{bmatrix}\)

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

To show : matrix AB = 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 AB = Matrix BA = \(\begin{bmatrix} 1 & 0 & 0 \\[0.3em] 0 & 1 & 0\\[0.3em] 0 & 0& 1 \end{bmatrix}\)

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