Matrices & Determinants
Adjoint
Grade 12
Question:
<p>If \(A\) is a \(3 \times 3\) matrix with \det(A)=a, then \det(adj(adj \(A)) equals:</p>
<p>(A) a</p>
<p>(B) \(a^2\)</p>
<p>(C) \(a^3\)</p>
<p>(D) \(a^4\)</p>
Step-by-Step Solution
Key Concept: For n \times n, $\text{adj}(\text{adj } $A)$ = (det $A)^(n-2) $A$ \Rightarrow determinant becomes (det $A)^(n-1). For 3 \times 3 \to a^3.
1. For $3 \times 3$: adj(adj $A) = (det $A) \cdot $A$
2. \det(adj(adj $A)) = (det $A)^3
3. = $a^3$
Correct Answer: 3