Let L, M, and N be non-singular matrices of order 3 satisfying the equations L2 = L-1, M = L8 and N = L2. Which ONE of the following is the value of the determinant of (M - N)? A) 0 B) 1 C) 2 D) 3

GATE CSE 2025 Set 2 · Q14

MCQ1 markEasy

Time00:00

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

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Solution

Official answer: A) 0

Linear Algebra

We are given that $L$ is an invertible matrix of order 3 satisfying:

L^2 = L^{-1}

Multiplying both sides on the right by $L$ :

L^3 = I

Now consider the matrix $M$ :

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

We are given $N = L^2$ , so:

M - N = L^2 - L^2 = O_{3 \times 3}

Since $M - N$ is the zero matrix of order 3, its determinant is:

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

Thus, (A) is the correct option.

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This Linear Algebra previous year question is Q14 of the GATE CSE 2025 Set 2 paper. Find more GATE CSE PYQs on Linear Algebra in Engineering Mathematics, or browse all GATE CSE PYQs.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.