Question

Let \( L \), \( M \), and \( N \) be non-singular matrices of size \( 3 \times 3 \), such that \( L^2 = L^{-1} \), \( M = L^8 \), and \( N = L^2 \). Find \( |M - N| \).

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

When dealing with matrix powers and inverses, it's often helpful to use properties like \( L^4 = I \) and matrix identities to simplify the calculations.

Answer 1

The Correct Option is A

Step 1: Using the given conditions. We are given that \( L^2 = L^{-1} \), so: \[ L^4 = I \quad \text{(since multiplying both sides by \( L^2 \))} \] Step 2: Calculate \( M - N \). From the given relations: \[ M = L^8 = (L^4)^2 = I^2 = I \] \[ N = L^2 \] Thus, \( M - N = I - L^2 \). Step 3: Finding the determinant. Since \( L^2 = L^{-1} \), we have: \[ I - L^2 = 0 \quad \text{(because \( L^2 \) is the inverse of \( L \))} \] Hence, the determinant of \( M - N \) is \( 0 \): \[ |M - N| = 0 \] Thus, the correct answer is \( 0 \).