Question:
For a \( 3 \times 3 \) non singular matrix A, if \( \text{Adj}(\text{Adj}(\text{Adj}(\text{Adj}(A)))) = |A|^n A \), then \( n = \)
For a \( 3 \times 3 \) non singular matrix A, if \( \text{Adj}(\text{Adj}(\text{Adj}(\text{Adj}(A)))) = |A|^n A \), then \( n = \)
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The general formula for \(k\) applications of adjoint is \( \text{Adj}^{(k)}(A) = |A|^{(m-1)^k / m} \times \dots \) which simplifies significantly for \(m=3\). Remember: \( \text{Adj}(\text{Adj}(A)) = |A|^{m-2} A \).
Updated On: Jun 9, 2026
- \( 3 \)
- \( 4 \)
- \( 8 \)
- \( 5 \)
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The Correct Option is A
Solution and Explanation
Concept:
For an \( m \times m \) matrix, the property holds: \( \text{Adj}(\text{Adj}(A)) = |A|^{m-2} A \).
Step 1: Apply the property for \( m=3 \).
\( \text{Adj}(\text{Adj}(A)) = |A|^{3-2} A = |A|A \).
Step 2: Iterate the application.
\( \text{Adj}(\text{Adj}(\text{Adj}(\text{Adj}(A)))) = \text{Adj}(\text{Adj}(|A|A)) = |A|^2 \text{Adj}(\text{Adj}(A)) = |A|^2 (|A|A) = |A|^3 A \).
Step 3: Compare with \( |A|^n A \).
Therefore, \( n = 3 \). 3
Step 1: Apply the property for \( m=3 \).
\( \text{Adj}(\text{Adj}(A)) = |A|^{3-2} A = |A|A \).
Step 2: Iterate the application.
\( \text{Adj}(\text{Adj}(\text{Adj}(\text{Adj}(A)))) = \text{Adj}(\text{Adj}(|A|A)) = |A|^2 \text{Adj}(\text{Adj}(A)) = |A|^2 (|A|A) = |A|^3 A \).
Step 3: Compare with \( |A|^n A \).
Therefore, \( n = 3 \). 3
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