A) There exist more than one but finite number of B's such that AB = BA
B) There exists exactly one B such that AB = BA
C) There exist infinitely many B's such that AB=BA
D) There cannot exist any B such that AB = BA
Correct Answer: C
Solution :
[c] \[AB=\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} a & 0 \\ 0 & b \\ \end{matrix} \right]=\left[ \begin{matrix} a & 2b \\ 3a & 4b \\ \end{matrix} \right]\] and \[BA=\left[ \begin{matrix} a & 0 \\ 0 & b \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \right]=\left[ \begin{matrix} a & 2a \\ 3b & 4b \\ \end{matrix} \right]\] If \[AB=BA\] \[\Rightarrow \left[ \begin{matrix} a & 2b \\ 3a & 4b \\ \end{matrix} \right]=\left[ \begin{matrix} a & 2a \\ 3b & 4b \\ \end{matrix} \right]\Rightarrow a=b\] From the above it is clear that there exist infinitely many B?s such that AB = BA.
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