Let L, M, and N be non-singular 3×3 matrices such that: L^2=L^−1,M=L^8,N=L^2…
2025
Let L, M, and N be non-singular 3×3 matrices such that:
L^2=L^−1,M=L^8,N=L^2
You have to find the value of the determinant of the matrix (M−N) and select the correct option from the given choices.
- A.
0
- B.
1
- C.
2
- D.
3
Attempted by 124 students.
Show answer & explanation
Correct answer: A
Key insight: use the relation L^2 = L^{-1} to reduce powers of L.
Step 1: From L^2 = L^{-1}, multiply both sides on the right by L to get L^3 = I.
Step 2: Reduce powers of L modulo 3. Since 8 ≡ 2 (mod 3), M = L^8 = L^2.
Step 3: Given N = L^2, we have M = L^2 and N = L^2, so M − N is the zero matrix.
Conclusion: The determinant of the zero 3×3 matrix is 0, so det(M − N) = 0.
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