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# Find the values of $x$ for which the given matrix $\left[ {\begin{array}{*{20}{c}} { - x}&x&2 \\ 2&x&{ - x} \\ x&{ - 2}&{ - x} \end{array}} \right]$ will be non-singular.A). $- 2 \leqslant x \leqslant 2$B). For all$x$other than $2$and $- 2$C). $x \geqslant 2$D). $x \leqslant - 2$

Last updated date: 22nd Jul 2024
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Hint: We need to find that for which values of $x$ the given matrix $\left[ {\begin{array}{*{20}{c}} { - x}&x&2 \\ 2&x&{ - x} \\ x&{ - 2}&{ - x} \end{array}} \right]$ will be non-singular. Non-singular matrix is a matrix whose determinant is non-zero. So we need to find the values of $x$ for which determinant of the above matrix will be a non-zero value.

Complete step-by-step solution:
We have to check that for which values of $x$ the given matrix $\left[ {\begin{array}{*{20}{c}} { - x}&x&2 \\ 2&x&{ - x} \\ x&{ - 2}&{ - x} \end{array}} \right]$ will be non-singular. Non-singular matrix is a matrix whose determinant is non-zero. So we need to check the values of $x$ for which determinant of the above matrix will be a non-zero value.
Let us calculate the determinant of the above matrix,
$\left| {\begin{array}{*{20}{c}} { - x}&x&2 \\ 2&x&{ - x} \\ x&{ - 2}&{ - x} \end{array}} \right|$
On applying the formula for the determinant we get,
$= - x( - {x^2} - 2x) - x( - 2x + {x^2}) + 2( - 4 - {x^2})$
On performing the multiplication of all the terms in the equation we get,
$= {x^3} + 2{x^2} - {x^3} + 2{x^2} - 8 - 2{x^2}$
On performing all the operations in the above equation we get,
$= 2{x^2} - 8$
Now, we need to find the condition that the matrix will be non-singular.
The condition for the matrix to be non-singular is, the determinant of the matrix should have non-zero value.
Thus,
$2{x^2} - 8 \ne 0$
Therefore,
$x \ne 2{\text{ and }}x \ne - 2$
This is a required condition for which the matrix will be non-singular.
Hence option B) For all $x$ other than $2$ and $- 2$ is correct.
As from the calculations, required condition for which matrix will be non-singular is $x \ne 2{\text{ and }}x \ne - 2$, hence option A) $- 2 \leqslant x \leqslant 2$ is incorrect.
As from the calculations, the required condition for which matrix will be non-singular is $x \ne 2{\text{ and }}x \ne - 2$, hence option C) $x \geqslant 2$ is incorrect.
As from the calculations, the required condition for which the matrix will be non-singular is $x \ne 2{\text{ and }}x \ne - 2$, hence option D) $x \leqslant - 2$ is incorrect.

Note: Singular matrix is a matrix whose determinant is zero and non-singular matrix is a matrix whose determinant is non-zero. If the condition is in the square form then we need to consider both positive and negative square root values of the solution.