Q47 / 160 · AP EAMCET 2021 (25 Aug Shift 1) · Mathematics

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

  • A
    $-2 \leq x \leq 2$
  • Correct answer
    B
    for all $x$ other than 2 and -2
  • C
    $x \geq 2$
  • D
    $x \leq-2$

Solution

Let $A=\left[\begin{array}{ccc}-x & x & 2 \\ 2 & x & -x \\ x & -2 & -2\end{array}\right]$ $A$ is non-singular matrix, $$ \begin{aligned} & \therefore|A| \neq 0 \\ & |A|=-x(-2 x-2 x)-x\left(-4+x^2\right)+2\left(-4-x^2\right) \neq 0 \\ & \Rightarrow \quad 4 x^2+4 x-x^3-8-2 x^2 \neq 0 \\ & \Rightarrow \quad x^3-2 x^2-4 x+8 \neq 0 \\ & \Rightarrow \quad(x-2)^2(x+2) \neq 0 \\ & \Rightarrow \quad x \neq-2,2 \end{aligned} $$

Hence, the correct answer is option: 'B'.

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