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Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]$ where $a, b$ are natural numbers, then which one of the following is correct?
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The correct answer is:
There exist infinitely many $B$ 's such that $A B=B A$
$A B=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]=\left[\begin{array}{cc}a & 2 b \\ 3 a & 4 b\end{array}\right]$
and $B A=\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]=\left[\begin{array}{cc}a & 2 a \\ 3 b & 4 b\end{array}\right]$
If $A B=B A$
$\Rightarrow\left[\begin{array}{cc}a & 2 b \\ 3 a & 4 b\end{array}\right]=\left[\begin{array}{cc}a & 2 a \\ 3 b & 4 b\end{array}\right] \Rightarrow a=b$
From the above it is clear that there exist infinitely many B's such that $\mathrm{AB}=\mathrm{BA}$.
and $B A=\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]=\left[\begin{array}{cc}a & 2 a \\ 3 b & 4 b\end{array}\right]$
If $A B=B A$
$\Rightarrow\left[\begin{array}{cc}a & 2 b \\ 3 a & 4 b\end{array}\right]=\left[\begin{array}{cc}a & 2 a \\ 3 b & 4 b\end{array}\right] \Rightarrow a=b$
From the above it is clear that there exist infinitely many B's such that $\mathrm{AB}=\mathrm{BA}$.
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