Mathematics - Matrices Question with Solution | TestHub

Given: $A=\left[\begin{array}{cc}\sqrt{2}& 1\\ -1& \sqrt{2}\end{array}\right]$

$\Rightarrow \left|A\right|=2+1=3$

$B=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\Rightarrow \left|B\right|=1$

And $C=AB{A}^T$

$\Rightarrow \left|C\right|=\left|AB{A}^T\right|$

$\Rightarrow \left|C\right|={\left|A\right|}^2\left|B\right| \left\{\text{as }\left|A\right|=\left|A^T\right|\right\}$

$\Rightarrow \left|C\right|=9$

 Now, solving $X=A{C}^2{A}^T$

.$\Rightarrow \left|X\right|=\left|A\right|\left|C^2\right|\left|A^T\right|$

$\Rightarrow \left|X\right|=3\times 9^2\times 3$

$\Rightarrow \left|X\right|=729$