Matrices & Determinants
Adjugate Properties
nta_pyq_2025_apr
Grade 12
Question:
For a $3 \times 3$ matrix $M$, let trace$(M)$ denote the sum of all diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace$(A) = 3$. If $B = \text{adj}(\text{adj}(2A))$, then the value of $|B| + $ trace$(B)$ equals:
Step-by-Step Solution
Key Concept: Use $\text{adj}(\text{adj}(M)) = |M|^{n-2} \cdot M$ for an $n \times n$ matrix. Here $n = 3$, so $\text{adj}(\text{adj}(2A)) = |2A|^1 \cdot 2A = 8|A| \cdot 2A = 8A$.
$B = \text{adj}(\text{adj}(2A)) = |2A| \cdot 2A = 8|A| \cdot 2A = 8A$. $\text{tr}(B) = 8 \cdot \text{tr}(A) = 24$. $|B| = 8^3 \cdot |A| = 512 \cdot \frac{1}{2} = 256$. $|B| + \text{tr}(B) = 256 + 24 = 280$.
Correct Answer: 280