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buckaroobill
04-16-2007, 01:29 PM
This was confusing me so any help would be appreciated!

Let M and N be n by n matrices, let n be odd, and suppose that MN = -NM. Prove that either M or N must be singular.

pka
04-16-2007, 01:58 PM
Suppose the neither M nor N is singular. Then
\L \begin{array}{l}
\left| M \right| \not= 0\quad \& \quad \left| N \right| \not= 0 \\
\left| {MN} \right| = \left| M \right|\left| N \right| \\
\left| { - NM} \right| = \left( { - 1} \right)^n \left| N \right|\left| M \right| = - \left| N \right|\left| M \right|,\quad {\rm{n is odd}}{\rm{.}} \\
\left| N \right|\left| M \right| = - \left| N \right|\left| M \right|\quad \Rightarrow \quad \left| N \right| = - \left| N \right| \\
\end{array}.

That is a contradiction. Thus either M or N is singular.