Q64 / 155 · WBJEE 2019 · Mathematics

If $A=\left(\begin{array}{ccc}5 & 5 x & x \\ 0 & x & 5 x \\ 0 & 0 & 5\end{array}\right)$ and $\left|A^{2}\right|=25,$ then $|x|$ is equal to

  • Correct answer
    A
    $\frac{1}{5}$
  • B
    5
  • C
    $5^{2}$
  • D
    1

Solution

Given $$ \begin{aligned} A &=\left[\begin{array}{ccc} 5 & 5 x & x \\ 0 & x & 5 x \\ 0 & 0 & 5 \end{array}\right] \\ \therefore \quad A^{2}=A \cdot A &=\left[\begin{array}{ccc} 5 & 5 x & x \\ 0 & x & 5 x \\ 0 & 0 & 5 \end{array}\right]\left[\begin{array}{ccc} 5 & 5 x & x \\ 0 & x & 5 x \\ 0 & 0 & 5 \end{array}\right] \\ &\left[\begin{array}{ccc} 25 & 25 x+5 x^{2} & 5 x+25 x^{2}+5 x \\ 0 & x^{2} & 5 x^{2}+25 \\ 0 & 0 & 25 \end{array}\right] \\ &=25\left(25 x^{2}-0\right) \\ &=25\left(25 x^{2}\right) \end{aligned} $$ since, given that $|A|^{2}=25$ Now, $25\left(25 x^{2}\right)=25$ $\begin{array}{lr}\Rightarrow & 25 x^{2}=1 \\ \Rightarrow & x^{2}=\frac{1}{25} \\ \Rightarrow & x=\pm \frac{1}{5}\end{array}$ $\Rightarrow \quad |x|=\frac{1}{5}$

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

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