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(B) Let $A = \left[\begin{array}{ccc}-x & x & 2 \ 2 & x & -x \ x & -2 & -2\end{array}ight]$.$A$ matrix $A$ is non-singular if and only if its dete

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for all $x$ other than $2$ and $-2$

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(B) Let $A = \left[\begin{array}{ccc}-x & x & 2 \ 2 & x & -x \ x & -2 & -2\end{array}ight]$.$A$ matrix $A$ is non-singular if and only if its determinant $|A| 
eq 0$.Calculating the determinant $|A|$:$|A| = -x \begin{vmatrix} x & -x \ -2 & -2 \end{vmatrix} - x \begin{vmatrix} 2 & -x \ x & -2 \end{vmatrix} + 2 \begin{vmatrix} 2 & x \ x & -2 \end{vmatrix}$$|A| = -x(-2x - 2x) - x(-4 + x^2) + 2(-4 - x^2)$$|A| = -x(-4x) + 4x - x^3 - 8 - 2x^2$$|A| = 4x^2 + 4x - x^3 - 8 - 2x^2$$|A| = -x^3 + 2x^2 + 4x - 8$For the matrix to be non-singular,$|A| 
eq 0$:$-x^3 + 2x^2 + 4x - 8 
eq 0$$-(x^3 - 2x^2 - 4x + 8) 
eq 0$$-(x^2(x - 2) - 4(x - 2)) 
eq 0$$-(x^2 - 4)(x - 2) 
eq 0$$-(x - 2)(x + 2)(x - 2) 
eq 0$$-(x - 2)^2(x + 2) 
eq 0$Therefore,$x 
eq 2$ and $x 
eq -2$.Thus,the matrix is non-singular for all $x$ other than $2$ and $-2$.

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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