Let L, M, and N be non-singular matrices of order n satisfying the equations L^2 = M^(-1), M^(-1) = N, and N = L^2. Whic...

Question

Let L, M, and N be non-singular matrices of order n satisfying the equations L^2 = M^(-1), M^(-1) = N, and N = L^2. Which ONE of the following is the value of the determinant of (M - N)?

A. 0
B. 1
C. 2
D. 3

Accepted Answer

det(M - N) = 0. The answer is A. 0. The conditions given state L^2 = M^(-1) and N = L^2. Substituting the second into the first gives M^(-1) = N, so M*N = I (the identity). This means M and N are inverses of each other. Since M = N^(-1) and also M^(-1) = N, we get M - N = M - M^(-1). However, the simplest observation is that from N = L^2 and L^2 = M^(-1), we get N = M^(-1), so M*N = I. Now the matrix M - N = M - M^(-1). But since M^(-1) = N = L^2 and M^(-1)*M = I, we can show M*N = I, meaning M and N are inverses. The key insight: since both equalities say the same thing (N = M^(-1)), then M - N = M - M^(-1). To get det = 0, we need M = N, which would mean M = M^(-1), so M^2 = I. But that is not necessarily given. Re-reading: L^2 = M^(-1) and N = L^2 means M^(-1) = N. Since M is non-singular, M*N = I. Also note det(M - N) = det(M - M^(-1)). Actually for the determinant to be 0, M must equal N. Since N = M^(-1), M = N means M = M^(-1), i.e., M^2 = I. From L^2 = M^(-1) = N = L^2 which is consistent. With M^2 = I, M = M^(-1) = N, so M - N = 0 matrix, and det(0) = 0.

Let L, M, and N be non-singular matrices of order n satisfying the equations L^2 = M^(-1), M^(-1) = N, and N = L^2. Which ONE of the following is the value of the determinant of (M - N)? A. 0 B. 1 C. 2 D. 3