Discussion
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
The determinant of (M - N) is 0.
From \(L^{2} = L^{-1}\), multiply by \(L\): \(L^{3} = I\). Then
\[ M = L^{8} = L^{6}L^{2} = (L^{3})^{2}L^{2} = L^{2} = N \]
So \(M - N\) is the zero matrix, and its determinant is 0.
The values 1, 2 and 3 would need M and N to differ. They cannot, because the cube of L is the identity, so every power of L repeats with period 3, and 8 and 2 leave the same remainder modulo 3.
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