Matrices & Determinants
Determinant Expression — adj and Scalar Multiples
nta_pyq_2024_apr
Grade 12
Question:
Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $|A^TA(\text{adj}(2A))^{-1}(\text{adj}(4B))(\text{adj}(AB))^{-1}AA^T|$ is equal to:
Step-by-Step Solution
Key Concept: Use $|\text{adj}(kA)|=k^{n(n-1)}|A|^{n-1}$ and $|(\text{adj}(M))^{-1}|=1/|\text{adj}M|$. For $n=3$: $|\text{adj}A|=|A|^2$, $|\text{adj}(2A)|=2^6|A|^2$, $|\text{adj}(4B)|=4^6|B|^2$, $|\text{adj}(AB)|=|AB|^2$.
Combining all determinant factors gives $64$.
Correct Answer: 4