Question-96

Let \(\displaystyle L,M,N\) be non-singular matrices of order \(\displaystyle 3\) satisfying the following equations:

\[ \begin{aligned} L^{2} & =L^{-1}\\ M & =L^{8}\\ N & =L^{2} \end{aligned} \]

Compute \(\displaystyle \text{det}( M-N)\).

NoteHint

Multiply both sides of the first equation by \(L\).

NoteAnswer

\(0\)

NoteSolution

We see that \(\displaystyle L^{3} =I\). With this, we can conclude the following:

\[ \begin{aligned} M & =L^{8}\\ & =\left( L^{3}\right)^{2} L^{2}\\ & =IL^{2}\\ & =L^{2} \end{aligned} \] Since \(\displaystyle M=L^{2}\) and \(\displaystyle N=L^{2}\), \(\displaystyle M=N\), and \(\displaystyle \text{det}( M-N) =0\).