Matrices & Determinants
Determinant expression from matrix equation
MJAT_TS8_P2
Grade 12
Question:
Let $A$ and $B$ be invertible matrices of order 3 with $|B|=2$ satisfying $A^T A B = ABB^T$. Find $|AB^{-1}\cdot\text{adj}(A^TB)^{-1}|$:
Step-by-Step Solution
Key Concept: From $A^TAB=ABB^T$: $|A|^3|A||B|=|A||B||B|^3\Rightarrow|A|^3=|B|^3\Rightarrow|A|=|B|=2$. Then $|AB^{-1}\cdot\text{adj}(A^TB)^{-1}|=|AB^{-1}|\cdot|\text{adj}(A^TB)|^{-1}$.
$\approx\mathbf{0.06}$.
Correct Answer: 0.06