Question

Let $A=\left(\begin{array}{cc}1& 2\\ & \\ 3& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}a& 0\\ & \\ 0& b\end{array}\right)$
where a,b $ϵ$ N. Then

• A. there cannot exist any B such that AB = BA.
• B. there exist more than one but finite number of B's such that AB = BA.
• C. there exists exactly one B such that AB = BA.
• D. there exist infinitely many B's such that AB = BA.

Answer 1

The correct option is D there exist infinitely many B's such that AB = BA.

$A=\left(\begin{array}{cc}1& 2\\ & \\ 3& 4\end{array}\right)$, $B=\left(\begin{array}{cc}a& 0\\ & \\ 0& b\end{array}\right)$
Now AB = BA $\Rightarrow$ when a = b