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The values of $x$ for which the given matrix $\left[\begin{array}{ccc}-x & x & 2 \\ 2 & x & -x \\ x & -2 & -2\end{array}\right]$ will be non-singular are
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Verified Answer
The correct answer is:
for all $x$ other than 2 and -2
Let $A=\left[\begin{array}{ccc}-x & x & 2 \\ 2 & x & -x \\ x & -2 & -2\end{array}\right]$
$A$ is non-singular matrix,
$$
\begin{aligned}
& \therefore|A| \neq 0 \\
& |A|=-x(-2 x-2 x)-x\left(-4+x^2\right)+2\left(-4-x^2\right) \neq 0 \\
& \Rightarrow \quad 4 x^2+4 x-x^3-8-2 x^2 \neq 0 \\
& \Rightarrow \quad x^3-2 x^2-4 x+8 \neq 0 \\
& \Rightarrow \quad(x-2)^2(x+2) \neq 0 \\
& \Rightarrow \quad x \neq-2,2
\end{aligned}
$$
$A$ is non-singular matrix,
$$
\begin{aligned}
& \therefore|A| \neq 0 \\
& |A|=-x(-2 x-2 x)-x\left(-4+x^2\right)+2\left(-4-x^2\right) \neq 0 \\
& \Rightarrow \quad 4 x^2+4 x-x^3-8-2 x^2 \neq 0 \\
& \Rightarrow \quad x^3-2 x^2-4 x+8 \neq 0 \\
& \Rightarrow \quad(x-2)^2(x+2) \neq 0 \\
& \Rightarrow \quad x \neq-2,2
\end{aligned}
$$
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