Computer Sciences > GATE 2025 SET-2 > Linear Algebra
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)?
L2=L-1, M=L8 and N=L2.
Which ONE of the following is the value of the determinant of (M-N)?
Correct : a
The correct answer is Option A - 0.
Given: L2 = L−1, M = L8, N = L2.
Order of L:
From L2 = L−1, multiply both sides by L:
L3 = L · L−1 = I (the identity matrix)
So L has order 3 — L3 = I.
Simplifying M = L8:
Since L3 = I, we reduce the exponent modulo 3:
8 = 3×2 + 2, so L8 = (L3)2 · L2 = I2 · L2 = L2
Computing M − N:
M = L8 = L2 = N
Therefore M − N = zero matrix
det(M − N) = det(0) = 0
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