Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Let $A$ be a $3 \times 3$ non-singular matrix then which of the following is not true
$|-A| = -|A|$
$|\text{Adj}(A)| = |A|^2$
$\text{Adj}(\text{Adj}(A)) = |A|^2 A$
$A \cdot \text{Adj}(A) = |A| I_3$
Step-by-Step Solution
Key Concept: For an n×n matrix, the adjugate of adjugate follows the formula adj(adj(A)) = |A|^(n-2)A. For n=3, this gives adj(adj(A)) = |A|^(3-2)A = |A|A, not |A|^2A as stated in option 3.
The adjugate formula gives $\text{adj}(\text{adj}A) = |A|^{n-2}A$ for an $n \times n$ matrix. Therefore $\text{adj}(\text{adj}A) = |A|^{2-2}A = |A|^0 A = A$.
Correct Answer: 3