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If $A=\left[\begin{array}{ccc}0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x\end{array}\right]$ is singular, then the possible values of $x$ are
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The correct answer is:
$0,12,-12$
We have, $A=\left[\begin{array}{ccc}0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x\end{array}\right]$
Since, $A$ is singular.
$$
\begin{aligned}
& &|A| &=0 \\
\Rightarrow & & & \\
\Rightarrow & x\left[x^{2}-0\right]+16[9 x-0] &=0 \\
\Rightarrow &-x^{3}+144 x &=0 \\
\Rightarrow & x\left(x^{2}-144\right) &=0 \\
\Rightarrow & x &=0 \\
& \Rightarrow & x &=\pm 12
\end{aligned}
$$
Since, $A$ is singular.
$$
\begin{aligned}
& &|A| &=0 \\
\Rightarrow & & & \\
\Rightarrow & x\left[x^{2}-0\right]+16[9 x-0] &=0 \\
\Rightarrow &-x^{3}+144 x &=0 \\
\Rightarrow & x\left(x^{2}-144\right) &=0 \\
\Rightarrow & x &=0 \\
& \Rightarrow & x &=\pm 12
\end{aligned}
$$
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