Matrices & Determinants
Nested adj Determinants — |3m+2n|
nta_pyq_2024_apr
Grade 12
Question:
Let $A$ be a non-singular matrix of order 3. If $\det(3\,\text{adj}(2\,\text{adj}((\det A)A)))=3^{-13}\cdot2^{-10}$ and $\det(3\,\text{adj}(2A))=2^m\cdot3^n$, then $|3m+2n|$ is equal to
Step-by-Step Solution
Key Concept: From the first equation, find $|A|$. Use $|\text{adj}(M)|=|M|^{n-1}$ for $n=3$. Expand step by step: $|(\det A)A|=|A|^4$, $|\text{adj}(|A|A)|=|A|^8$, $|2\,\text{adj}(|A|A)|=2^6|A|^8\cdot|A|^4$...
$|A|=2^{-1}3^{-1}$. $m=-4$, $n=-1$. $|3m+2n|=14$.
Correct Answer: 14