Matrices & Determinants
Determinant of Matrix Product — det(X)
nta_pyq_2024_jan
Grade 12
Question:
If $A=\begin{bmatrix}\sqrt{2}&1\\-1&\sqrt{2}\end{bmatrix}$, $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, $C=ABA^T$ and $X=A^TC^2A$, then $\det X$ is equal to:
Step-by-Step Solution
Key Concept: $\det(A)=3$, $\det(B)=1$. $\det(C)=\det(A)^2\det(B)=9$. $\det(X)=\det(A^T)\det(C)^2\det(A)=\det(A)^2\det(C)^2=9\times81=729$.
$\det A=3$, $\det B=1$, $|C|=9$. $|X|=|A|^2|C|^2=9\times81=729$.
Correct Answer: 2