If $\mathrm{A}=\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]$ and…
If $\mathrm{A}=\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}x & 0 \\ 0 & y\end{array}\right], \mathrm{x}, \mathrm{y} \in \mathrm{N}$, then
- There is exactly one such matrix $B$ such that $A B=I$
- There is no matrix $\mathrm{B}$ such that $\mathrm{AB}=\mathrm{BA}$
- There exist only a finite number of matrices $B$ such that $\mathrm{AB}=\mathrm{BA}$
- There exist infinite number of matrices $B$ such that $\mathrm{AB}=\mathrm{BA}$
Solution
We have
$
\begin{aligned}
& \mathrm{A}=\left[\begin{array}{ll}
3 & 4 \\
5 & 6
\end{array}\right] \mathrm{B}=\left[\begin{array}{ll}
\mathrm{x} & 0 \\
0 & \mathrm{y}
\end{array}\right] \mathrm{x}, \mathrm{y} \varepsilon \mathrm{n} \\
& \Rightarrow \mathrm{AB}=\left[\begin{array}{ll}
3 & 4 \\
5 & 6
\end{array}\right]\left[\begin{array}{ll}
\mathrm{x} & 0 \\
0 & \mathrm{y}
\end{array}\right] \\
& =\mathrm{AB}=\left[\begin{array}{ll}
3 \mathrm{x} & 4 \mathrm{y} \\
5 \mathrm{x} & 6 \mathrm{y}
\end{array}\right] \\
& =\mathrm{xy}\left[\begin{array}{ll}
3 \mathrm{x} & 4 \mathrm{y} \\
5 \mathrm{x} & 6 \mathrm{y}
\end{array}\right] \\
& \text { now } \mathrm{BA}=\mathrm{xy}\left[\begin{array}{ll}
3 & 4 \\
5 & 6
\end{array}\right] \\
& \text { Thus } \mathrm{AB}=\mathrm{BA}
\end{aligned}
$
Asked in: AP EAMCET 2022 (06 Jul Shift 1)
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