Matrices & Determinants
Nested adj — Finding m+2n
nta_pyq_2024_apr
Grade 12
Question:
If $A$ is a square matrix of order 3 such that $\det(A)=3$ and $\det(\text{adj}(-4\,\text{adj}(-3\,\text{adj}(3\,\text{adj}((2A)^{-1})))))=2^m3^n$, then $m+2n$ is equal to:
Step-by-Step Solution
Key Concept: Work from inside out using $|\text{adj}(kM)|=k^{n(n-1)}|M|^{n-1}$ for $n\times n$ matrix. Here $n=3$, so $|\text{adj}(M)|=|M|^2$ and scalar $k$ contributes $k^6$ to the determinant of the adjugate.
$m=-36$, $n=20$. $m+2n=4$.
Correct Answer: 4