Question

If $A$ is a square matrix of order 3 such that $|A| = 2$, then $|\operatorname{adj}(\operatorname{adj}(A))|$ is:

• A. 2
• B. 16
• C. -16
• D.

Hint:

Use the formula recursively for \(\operatorname{adj}(\operatorname{adj}(A))\) using the order of the matrix.

Answer 1

The Correct Option is C

Step 1: Recall that for an \(n \times n\) matrix: \[ |\operatorname{adj}(A)| = |\det(A)|^{n-1} \] Given \(n = 3\) and \(|A| = 2\), then: \[ |\operatorname{adj}(A)| = 2^{3-1} = 2^{2} = 4 \] Step 2: Find \(|\operatorname{adj}(\operatorname{adj}(A))|\). Since \(\operatorname{adj}(A)\) is also a \(3 \times 3\) matrix, apply the same formula: \[ |\operatorname{adj}(\operatorname{adj}(A))| = |\det(\operatorname{adj}(A))|^{3-1} = |\det(\operatorname{adj}(A))|^{2} \] Step 3: But \(\det(\operatorname{adj}(A)) = |\operatorname{adj}(A)|\), so: \[ \det(\operatorname{adj}(A)) = 4 \] Step 4: Substitute: \[ |\operatorname{adj}(\operatorname{adj}(A))| = 4^{2} = 16 \]