If A = [ cases 2 & 1 - 1 & 2 cases ], B[ cases 1 & 0 1 & 1 cases ],...

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

$⇒ \left|A\right| = 2 + 1 = 3$

B = \left[\begin{cases}1 & 0 \\ 1 & 1\end{cases}\right] ⇒ \left|B\right| = 1

And $C = AB{A}^{T}$

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

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

$⇒ \left|C\right| = 9$

 Now, solving $X = A{C}^{2}{A}^{T}$

.$⇒ \left|X\right| = \left|A\right|\left|{C}^{2}\right|\left|{A}^{T}\right|$

$⇒ \left|X\right| = 3 \times {9}^{2} \times 3$

$⇒ \left|X\right| = 729$