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

Question

Compute AB and BA, which ever exists when

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

A = [(2,-1)(3,0)(-1,4)]

B = [(2,3)(0,4)]

Answer 1

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

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

To find : matrix 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 b and B is a matrix of order c 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 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 = 2,d = 2 ,thus matrix AB is of order 3 x 2

For matrix BA, a = 3,b = c = 2,d = 2 ,thus matrix BA exists, if and only if d = a

But 3 ≠ 2

Thus matrix BA does not exist