Question

Let \( L, M, \) and \( N \) be non-singular matrices of order 3 satisfying the equations: \[ L^2 = L^{-1}, \quad M = L^8, \quad N = L^2. \] Which ONE of the following is the value of the determinant of \( (M - N) \)?

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

For matrices satisfying power equations, use determinant properties such as \( \det(A \cdot B) = \det(A) \cdot \det(B) \) to simplify calculations in GATE problems.

Answer 1

The Correct Option is A

Given the matrix equation: \[ L^2 = L^{-1} \] Taking determinants on both sides: \[ \det(L^2) = \det(L^{-1}) \] Since \( \det(L^{-1}) = \frac{1}{\det(L)} \), we get: \[ (\det L)^2 = \frac{1}{\det L} \] Let \( x = \det L \), then: \[ x^3 = 1 \Rightarrow x = 1 { (since } L { is non-singular, } x \neq 0 {)} \] Now, using \( M = L^8 \) and \( N = L^2 \): \[ \det(M) = \det(L^8) = (\det L)^8 = 1^8 = 1 \] \[ \det(N) = \det(L^2) = (\det L)^2 = 1^2 = 1 \] Thus, \[ \det(M - N) = \det(1 - 1) = \det(0) = 0. \] Therefore, the correct answer is option (A).