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Let A be a 3x3 matrix with elements such that tr(A) = (l - 3) + 6 + (9 - l) = 12 (the diagonal entries are l-3, 6, and 9-l). Let B = adj(A) and C = adj(B). If |A| = 5, find tr(C).

1. 5
2. 12
3. 30
4. 60

Correct answer: 60

Solution

For a 3x3 matrix A, the property adj(adj(A)) = |A|^(n-2) * A = |A|¹ * A holds. The trace of A is tr(A) = (l-3) + 6 + (9-l) = 12 (the l terms cancel). Since B = adj(A) and C = adj(B) = adj(adj(A)) = |A| * A, we get tr(C) = |A| * tr(A) = 5 * 12 = 60.

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