If A = [{√2, 1}, {-1, √2}], B = [{1, 0}, {1, 1}], C = ABA^T and X = A^TC^2A,

Question

If \(\mathrm{A}=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], \mathrm{C}=\mathrm{ABA}^{\mathrm{T}}\) and \(\mathrm{X}=\mathrm{A}^{\mathrm{T}} \mathrm{C}^{2} \mathrm{~A}\), then \(\operatorname{det} \mathrm{X}\) is equal to :

(1) 243

(2) 729

(3) 27

(4) 891

Answer 1

Correct option is (2) 729

\(A=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right] \Rightarrow \operatorname{det}(A)=3\)

\(B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right] \Rightarrow \operatorname{det}(B)=1\)

Now 

\(\mathrm{C}=\mathrm{ABA}^{\mathrm{T}} \Rightarrow \operatorname{det}(\mathrm{C})=(\operatorname{det}(\mathrm{A}))^{2} \mathrm{x} \operatorname{det}(\mathrm{B})\)

\(|C|=9\)

Now

\(|\mathrm{X}|=\left|\mathrm{A}^{\mathrm{T}} \mathrm{C}^{2} \mathrm{~A}\right|\)

\(=\left|\mathrm{A}^{\mathrm{T}}\right||\mathrm{C}|^{2}|\mathrm{~A}|\)

\(=|\mathrm{A}|^{2}|\mathrm{C}|^{2}\)

\(=9 \times 81\)

\(=729\)