For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) : A = [(1,2,-3)(4,-5,6)] B = [(3,-4)(2,1)(-1,0)]

Question

For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :

A = \( \begin{bmatrix} -1& 2 &-3 \\[0.3em] 4 &-5 & 6 \\[0.3em] \end{bmatrix}\) and B = \( \begin{bmatrix} 3& -4 \\[0.3em] 2 &1 \\[0.3em] -1 &0 \end{bmatrix}\) 

A = [(1,2,-3)(4,-5,6)]

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

Answer 1

Let us take C = AB

LHS = C’

\(\Rightarrow\) \( \begin{bmatrix} 4 & -4 \\[0.3em] 6&-21 \\[0.3em] \end{bmatrix}\)

A' = \( \begin{bmatrix} -1 & 4 \\[0.3em] 2&-21 \\[0.3em] -3 &6 \end{bmatrix}\) 

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

RHS = B’A’

LHS = RHS

Hence proved.