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Let A = [a_ij] be a 3x3 matrix such that A*P = 2I, where I is the 3x3 identity matrix and P = [[1,2,3],[0,1,4],[0,0,1]]. Which of the following hold? (Note: adj(M) denotes the adjoint and Tr(M) denotes the trace of matrix M.)
- Tr(adj(adj(A))) = 48
- Tr(adj(adj(A))) = 12
- det(adj(A) * adj(adj(A))) = 2¹⁸
- det(adj(A) * adj(adj(A))) = 2²⁴
Correct answer: Tr(adj(adj(A))) = 48
Solution
A = 2*P^(-1). Since det(P) = 1, det(A) = 8. Tr(A) = 2*Tr(P^(-1)) = 2*3 = 6. adj(adj(A)) = det(A)*A, so Tr(adj(adj(A))) = det(A)*Tr(A) = 8*6 = 48. Also det(adj(A)) = det(A)² = 64, det(adj(adj(A))) = det(A)³ * det(A) = 8⁴ = 2¹², so det(adj(A)*adj(adj(A))) = 64*2¹² = 2¹⁸.
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