GATE CS 2025 Set 2 Q.14: Linear Algebra

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 $(M - N)$ ?

$0$

$1$

$2$

$3$

Official answer and solution

Solution

Multiply $L^2 = L^{-1}$ by $L$ to get $L^3 = I$ , simplify $M = L^8$ , and compute $M - N$ .

Since $L$ is non-singular and $L^2 = L^{-1}$ , multiplying both sides by $L$ yields $L^3 = I$ . Therefore:

M = L^8 = (L^3)^2 L^2 = I^2 L^2 = L^2 = N

Consequently, $M - N = O$ (the $3 \times 3$ zero matrix), so $\det(M - N) = 0$ .

Final answer

Option (A)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 14 of 65
Section: Computer Science and Information Technology

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