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Let M and N be two non-singular real matrices of order 3 satisfying adj(M) = 2N and adj(N) = M. Which of the following are correct?

1. adj(M²*N) + adj(M*N²) = 4(M + 2N)
2. M = 2*N^(-1)
3. M*N = 4I
4. adj(M*N^(-1)) = 4*M^(-2)

Correct answer: adj(M²*N) + adj(M*N²) = 4(M + 2N)

Solution

From adj(M) = 2N and adj(N) = M one derives |M| = 4, |N| = 2, M^(-1) = N/2, N^(-1) = M/2, and MN = 2I. Checking each option: (A) evaluates to 4(M+2N) [correct]; (B) M = 2N^(-1) = 2*(M/2) = M [true]; (C) MN = 2I, not 4I [false]; (D) adj(MN^(-1)) = N² = 4M^(-2) [true]. So A, B, D are correct.

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