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If A = \(\begin{bmatrix} 2 & -3 & -5 \\[0.3em] -1& 4 &5\\[0.3em] 1 & -3&-4 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & -2 & -4 \\[0.3em] -1& 3 &4\\[0.3em] 1 & -2&-3 \end{bmatrix},\) shown that AB = A and BA = B.

A = [(2,-3,-5)(-1,4,5)(1,-3,-4)]

B = [(2,-2,-4)(-1,3,4)(1,-2,-3)]

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

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

To show : matrix AB = A, BA = B

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 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

Matrix AB = Matrix A

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

MATRIX AB = A and MATRIX BA = B

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