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 that \( \text{adj}(\text{adj}(A)) = A \). To solve this, we need to use the following property of the adjugate of a matrix: For any square matrix \( A \) of order \( n \), we have the relationship: \[ \text{adj}(A) = |A|^{n-1} A^{-1} \] Now, applying this property for the adjugate of the adjugate: \[ \text{adj}(\text{adj}(A)) = |A|^{n-1} \cdot \text{adj}(A)^{-1} \] Substitute \( \text{adj}(A) = |A|^{n-1} A^{-1} \) into the above equation: \[ \text{adj}(\text{adj}(A)) = |A|^{n-1} \cdot |A|^{n-1} A^{-1} \] \[ \text{adj}(\text{adj}(A)) = |A|^{2n-2} A^{-1} \] Given that \( \text{adj}(\text{adj}(A)) = A \), equate this expression to \( A \): \[ |A|^{2n-2} A^{-1} = A \] Multiply both sides by \( A \): \[ |A|^{2n-2} = |A|^2 \] Thus, \( |A|^{2n-2} = |A|^2 \). From this equation, it follows that: \[ |A| = 1 \] Thus, the determinant of matrix \( A \) is 1.