M and N are real unequal n x n matrices satisfying M^{3} = N^{3} and M^{2}N = N^{2}M. Can we choose M and N so that M^{2} + N^{2} is invertible?

**Solution**

Answer: No.

Easy.

The only difficulty is that we are not told whether or not it is true. So one has to (1) try experimenting a little to find invertible M, N satisfying the conditions, and (2) try proving you cannot find such M, N. If you have bad luck you will try in that order!

(M^{2} + N^{2})M = M^{3} + N^{2}M = N^{3} + M^{2}N = (M^{2} + N^{2})N. But now if M^{2} + N^{2} was invertible, we could multiply by its inverse to get M = N, whereas we are told M and N are unequal.

© John Scholes

jscholes@kalva.demon.co.uk

21 Sep 1999