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 \((M - N)\)?
Answer
(A)
Explanation
Multiply \(L^{2} = L^{-1}\) on the right by \(L\) to get \(L^{3} = I\). Powers of \(L\) therefore repeat every three steps:
\[M = L^{8} = (L^{3})^{2}L^{2} = L^{2} = N.\]
So \(M - N\) is the zero matrix, and its determinant is 0. Non-singularity only guarantees that \(L^{-1}\) exists; it says nothing about the difference \(M - N\), so the non-zero options cannot occur.
Answer: A