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$\left[\begin{array}{ll}0 & \mathrm{a} \\ \mathrm{b} & 0\end{array}\right]^{4}=1$, then
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Solution:
2874 Upvotes
Verified Answer
The correct answer is:
$a b=1$
$\begin{array}{l}
\text { Here, }\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{2}=\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{2} \\
=\left[\begin{array}{ll}
0+a b & 0+0 \\
0+0 & a b+0
\end{array}\right]=\left[\begin{array}{ll}
a b & 0 \\
0 & a b
\end{array}\right]
\end{array}$
$\begin{array}{l}
\text { Similarly, }\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{4}=\left[\begin{array}{ll}
a b & 0 \\
0 & a b
\end{array}\right]\left[\begin{array}{ll}
a b & 0 \\
0 & a b
\end{array}\right] \\
=\left[\begin{array}{ll}
a^{2} b^{2}+0 & 0+0 \\
0+0 & 0+a^{2} b^{2}
\end{array}\right]=\left[\begin{array}{ll}
a^{2} b^{2} & 0 \\
0 & a^{2} b^{2}
\end{array}\right]
\end{array}$
Now,
$\begin{array}{l}
{\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{4}=I \Rightarrow\left[\begin{array}{ll}
a^{2} b^{2} & 0 \\
0 & a^{2} b^{2}
\end{array}\right]=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]} \\
\Rightarrow a^{2} b^{2}=1 \Rightarrow a b=1
\end{array}$
\text { Here, }\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{2}=\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{2} \\
=\left[\begin{array}{ll}
0+a b & 0+0 \\
0+0 & a b+0
\end{array}\right]=\left[\begin{array}{ll}
a b & 0 \\
0 & a b
\end{array}\right]
\end{array}$
$\begin{array}{l}
\text { Similarly, }\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{4}=\left[\begin{array}{ll}
a b & 0 \\
0 & a b
\end{array}\right]\left[\begin{array}{ll}
a b & 0 \\
0 & a b
\end{array}\right] \\
=\left[\begin{array}{ll}
a^{2} b^{2}+0 & 0+0 \\
0+0 & 0+a^{2} b^{2}
\end{array}\right]=\left[\begin{array}{ll}
a^{2} b^{2} & 0 \\
0 & a^{2} b^{2}
\end{array}\right]
\end{array}$
Now,
$\begin{array}{l}
{\left[\begin{array}{ll}
0 & a \\
b & 0
\end{array}\right]^{4}=I \Rightarrow\left[\begin{array}{ll}
a^{2} b^{2} & 0 \\
0 & a^{2} b^{2}
\end{array}\right]=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]} \\
\Rightarrow a^{2} b^{2}=1 \Rightarrow a b=1
\end{array}$
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