Question
Let \[A = \left( \begin{array}{l}
1\,\,\,\,\,\,2\\
3\,\,\,\,\,4
\end{array} \right){\rm{and }}\,\,B = \left( \begin{array}{l}
a\,\,\,\,\,\,0\\
0\,\,\,\,\,\,b
\end{array} \right),a,b \in N.\] Then
A.
there cannot exist any $$B$$ such that $$AB = BA$$
B.
there exist more then 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 :
there exist infinitely many $$B’s$$ such that $$AB = BA$$
Solution :
\[A = \left[ \begin{array}{l}
1\,\,\,\,\,\,2\\
3\,\,\,\,\,4
\end{array} \right]\,\,\,\,\,{\rm{ }}B = \left[ \begin{array}{l}
a\,\,\,\,\,\,0\\
0\,\,\,\,\,\,b
\end{array} \right]\]
\[AB = \left[ \begin{array}{l}
\,a\,\,\,\,\,\,\,2b\\
3a\,\,\,\,\,\,4b
\end{array} \right]\]
\[BA = \left[ \begin{array}{l}
a\,\,\,\,\,0\\
0\,\,\,\,\,b
\end{array} \right]\left[ \begin{array}{l}
1\,\,\,\,\,\,\,2\\
3\,\,\,\,\,\,4
\end{array} \right] = \left[ \begin{array}{l}
\,a\,\,\,\,\,\,\,2b\\
3a\,\,\,\,\,4b
\end{array} \right]\]
Hence, $$AB = BA$$ only when $$a = b$$
∴ There can be infinitely many $$B’s$$ for which $$AB = BA$$