Question

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). $x \leqslant - 2$

Answer 1

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,

{ - 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.