Compute AB and BA, which ever exists when A = [(0,1,-5)(2,4,-5)] and B = [(1,3)(-1,0)(0,5)]

Question

Compute AB and BA, which ever exists when

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

A = [(0,1,-5)(2,4,-5)]

B = [(1,3)(-1,0)(0,5)]

Answer 1

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

Matrix A is of order 2 x 3 and Matrix B is of order 3 x 2

To find : matrices AB and BA

Formula used :

Where c_ij = a_i1b_1j + a_i2b_2j + a_i3b_3j + ……………… + a_inb_nj

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 , f and only if d = a

For matrix AB, a = 2,b = 3,c = 3,d = 2 ,matrix AB exists and is of order 2 x 2, as

b = c = 3

For matrix BA, a = 2,b = 3,c = 3,d = 2 ,matrix BA exists and is of order 3 x 3, as d = a = 2