GATE CS 2025 Set 2 Q4 — Linear Algebra: Matrices, Determinants, System of Linear Equations

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 deteminant of

(M-N)

View worked solution

Answer: A

Given the equation for the non-singular matrix $L$ of order $3$ :

L^2 = L^{-1}

Since $L$ is non-singular, we can multiply both sides by $L$ :

L^3 = I

where $I$ is the identity matrix of order $3$ .

Now, let's simplify the expression for $M$ :

M = L^8 = (L^3)^2 \cdot L^2

Substituting $L^3 = I$ into the equation:

M = I^2 \cdot L^2 = L^2

We are also given:

N = L^2

Comparing $M$ and $N$ , we get:

M = N \implies M - N = O

where $O$ is the zero matrix of order $3$ .

Therefore, the determinant of $(M-N)$ is:

\det(M-N) = \det(O) = 0

Thus, the correct option is A.

Free practice · Your answers on this page are not saved.

Published 10 Oct 2026

Related questions