Question

If $ A $ is a square matrix such that $ \text{adj}(\text{adj}(A)) = A $, then $ |A| $ is:

• A. 1
• B. 3
• C. 0
• D. 9
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Hint:

When solving for the determinant of a matrix using the adjugate property, always remember that the adjugate of the adjugate of a matrix follows a specific relation to the original matrix's determinant.

Answer 1

The Correct Option is A

We are given the condition \( \text{adj}(\text{adj}(A)) = A \). The solution requires the property of the adjugate matrix: For any square matrix \( A \) of order \( n \), the relationship is: \[\text{adj}(A) = |A|^{n-1} A^{-1}\]Applying this property to the adjugate of the adjugate yields:\[\text{adj}(\text{adj}(A)) = |A|^{n-1} \cdot \text{adj}(A)^{-1}\]Substituting \( \text{adj}(A) = |A|^{n-1} A^{-1} \) into the equation above:\[\text{adj}(\text{adj}(A)) = |A|^{n-1} \cdot |A|^{n-1} A^{-1}\]\[\text{adj}(\text{adj}(A)) = |A|^{2n-2} A^{-1}\]Given \( \text{adj}(\text{adj}(A)) = A \), we equate this to \( A \):\[|A|^{2n-2} A^{-1} = A\]Multiplying both sides by \( A \):\[|A|^{2n-2} = |A|^2\]From \( |A|^{2n-2} = |A|^2 \), we deduce:\[|A| = 1\]Therefore, the determinant of matrix \( A \) is 1.