Matrices & Determinants
Adjoint of Adjoint — Determinant Properties
nta_pyq_2026_jan
Grade 12
Question:
Let $|A|=6$, where $A$ is a $3\times3$ matrix. If $|\text{adj}(3\,\text{adj}(A^2\cdot\text{adj}(2A)))|=2^m\cdot3^n$, $m,n\in\mathbb{N}$, then $m+n$ is equal to _____.
Step-by-Step Solution
Key Concept: $|kM|=k^3|M|$ for $3\times3$. $|\text{adj}(M)|=|M|^2$ for $3\times3$. $|2A|=8\cdot6=48$. $|\text{adj}(2A)|=48^2=2^8\cdot3^2$. $|A^2\cdot\text{adj}(2A)|=|A|^2\cdot|\text{adj}(2A)|=36\cdot2^8\cdot3^2=2^{10}\cdot3^4$.
$m+n=62$.
Correct Answer: 62