Product of matrices

Question

A= [ (2 −6) (-1  3)]  find B_2x2≠ ∅ and B_2x2≠Identity such that A and B commute

   

Answer 1

A = \(\begin{bmatrix}2&-6\\-1&3\end{bmatrix}\)

Let B = \(\begin{bmatrix}1&6\\1&0\end{bmatrix}\)

Then AB  = \(\begin{bmatrix}2&-6\\-1&3\end{bmatrix}\) \(\begin{bmatrix}1&6\\1&0\end{bmatrix}\)

 = \(\begin{bmatrix}-4&12\\2&-6\end{bmatrix}\)

BA = \(\begin{bmatrix}1&6\\1&0\end{bmatrix}\) \(\begin{bmatrix}2&-6\\-1&3\end{bmatrix}\)

 = \(\begin{bmatrix}-4&12\\2&-6\end{bmatrix}\)

\(\therefore\) AB = BA, where B = \(\begin{bmatrix}1&6\\1&0\end{bmatrix}\) \(\neq\) O, I