Matrices & Determinants
Matrices
nta_pyq_2025_jan
Grade 12
Question:
For a $3\times 3$ matrix $M$, let $\operatorname{trace}(M)$ denote the sum of all diagonal elements of $M$. Let $A$ be a $3\times 3$ matrix such that $|A|=\dfrac{1}{2}$ and $\operatorname{trace}(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2A))$, then the value of $|B|+\operatorname{trace}(B)$ equals:
Step-by-Step Solution
Key Concept: For $n\times n$: $\operatorname{adj}(\operatorname{adj}(M))=|M|^{n-2}M$. For $n=3$: $\operatorname{adj}(\operatorname{adj}(M))=|M|\,M.$ Also $|kM|=k^{n}|M|$ and $\operatorname{trace}(kM)=k\operatorname{trace}(M).$
$B=\operatorname{adj}(\operatorname{adj}(2A))=|2A|\cdot 2A=2^{3}|A|\cdot 2A=8\cdot\tfrac{1}{2}\cdot 2A=8A.$
$\operatorname{trace}(B)=8\operatorname{trace}(A)=24.$
$|B|=|8A|=8^{3}|A|=512\cdot\tfrac{1}{2}=256.$
$|B|+\operatorname{trace}(B)=256+24=280.$
Correct Answer: 4