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Let B be a 3x3 matrix with adj(B) = A. Let M and N be 3x3 matrices with det(M) = 1 = det(N). Consider the following: Statement I: adj(N^(-1) * B * M^(-1)) = M * A * N. Statement II: If P is a non-singular 3x3 matrix, then adj(P^(-1)) = (adj P)^(-1).

1. Statement I is true, Statement II is false.
2. Statement I is false, Statement II is true.
3. Statement I is true, Statement II is true.
4. Statement I is false, Statement II is false.

Correct answer: Statement I is true, Statement II is true.

Solution

Statement I: Let X = N^(-1)*B*M^(-1). det(X) = det(N^(-1))*det(B)*det(M^(-1)) = det(B). adj(X) = det(X)*X^(-1) = det(B)*(N^(-1)*B*M^(-1))^(-1) = det(B)*M*B^(-1)*N. Since adj(B)=A and B is 3x3: B^(-1) = adj(B)/det(B) = A/det(B). So adj(X) = det(B)*M*(A/det(B))*N = M*A*N. Statement I is TRUE. Statement II: adj(P^(-1)) = det(P^(-1))*P = P/det(P). And (adj P)^(-1) = (det(P)*P^(-1))^(-1) = P/det(P). These are equal. Statement II is TRUE.

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