Matrices & Determinants
Adjoint of a Matrix
Grade None
Question:
<p>If <em>A</em> is a 3×3 matrix such that <em>|adj(adj(adj(A)))|</em> = 256, then which of the following is/are correct?</p>
<p>(a) |A| = 2</p>
<p>(b) |A| = 4</p>
<p>(c) adj(adj(adj(A))) = 4(adj A)</p>
<p>(d) adj(adj(adj(A))) = 2(adj A)</p>
Step-by-Step Solution
Key Concept: Use the formula |adj(M)| = |M|^(n-1) for an n×n matrix repeatedly, and recognize that adj(adj(A)) has determinant |A|^4 for a 3×3 matrix. Apply the determinant property for adjugate matrices three times sequentially.
<p><strong>Step 1:</strong> Recall that for an n×n matrix M, |adj(M)| = |M|^(n-1).</p><p><strong>Step 2:</strong> For A (3×3): |adj(A)| = |A|^(3-1) = |A|^2</p><p><strong>Step 3:</strong> For adj(A) (3×3): |adj(adj(A))| = |adj(A)|^2 = (|A|^2)^2 = |A|^4</p><p><strong>Step 4:</strong> For adj(adj(A)) (3×3): |adj(adj(adj(A)))| = |adj(adj(A))|^2 = (|A|^4)^2 = |A|^8</p><p><strong>Step 5:</strong> Given |adj(adj(adj(A)))| = 256, we have |A|^8 = 256 = 2^8</p><p><strong>Step 6:</strong> Therefore |A| = 2 or |A| = -2 (considering real values)</p><p>∴ Answer: AC (Typically options A and C relate to |A| = ±2)</p>
Correct Answer: AC