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 8's such that\[AB=BA\]
D) there cannot exist any B such that\[AB=BA\]
Correct Answer: C
Solution :
Since, \[A=\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \right]\]and\[B=\left[ \begin{matrix} a & 0 \\ 0 & b \\ \end{matrix} \right]\] Now, \[AB=\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} a & 0 \\ 0 & b \\ \end{matrix} \right]=\left[ \begin{matrix} a & 2b \\ 3a & 4b \\ \end{matrix} \right]\] 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 \[AB=BA\Rightarrow a=b\] Hence, \[AB=BA\] is possible for infinitely many \[B's\].
You need to login to perform this action.
You will be redirected in
3 sec
