GATE CSE 2025 Session 2 Q14: Determinant of L^8 − L^2 when L^2 = L^(−1)

Let $L$ , $M$ , and $N$ be non-singular matrices of order 3 satisfying the equations $L^2 = L^{-1}$ , $M = L^8$ and $N = L^2$ .

Which ONE of the following is the value of the determinant of $(M - N)$ ?

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Answer: A (official GATE 2025 final answer key)

Concept used: Reduce matrix powers using the given relation.

First step: $L^2 = L^{-1}$ gives $L^3 = I$ , so $L^8 = L^6 L^2 = L^2$ and $M = N$ .

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Source: official GATE 2025 Session 2 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key