Question

Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\text{and}B=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right],a,b,ϵN.$ Then

• A. there cannot exists any B such that AB = BA.
• B. there exists 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 exists infinitely many B's such that AB = BA.

Answer 1

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

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]\therefore AB=\left[\begin{array}{cc}a& 2b\\ 3a& 4b\end{array}\right]andBA=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}a& 2a\\ 3b& 4b\end{array}\right]$
AB = BA only when a = b.