Matrices & Determinants
Determinant of adjoint matrix
nta_pyq_2023_jan
Grade 12
Question:
Let A be a $n \times n$ matrix such that $|A| = 2$. If the determinant of the matrix $\text{Adj}(2 \cdot \text{Adj}(2A^{-1}))$ is $2^{84}$, then $n$ is equal to ___.
Step-by-Step Solution
Key Concept: Use $|\text{Adj}(M)| = |M|^{n-1}$ repeatedly, and $|2A^{-1}| = \frac{2^n}{|A|}$
$|2A^{-1}| = 2^n/2 = 2^{n-1}$. $|\text{Adj}(2A^{-1})| = (2^{n-1})^{n-1} = 2^{(n-1)^2}$. $|2\,\text{Adj}(2A^{-1})| = 2^n \cdot 2^{(n-1)^2} = 2^{n+(n-1)^2}$. $|\text{Adj}(2\,\text{Adj}(2A^{-1}))| = (2^{n+(n-1)^2})^{n-1} = 2^{84}$. So $(n-1)(n+(n-1)^2) = 84$. Testing $n=5$: $4 \cdot (5+16) = 84$. Yes! Answer: 5
Correct Answer: 5