If A and B are n by n, AB = -BA, and n is odd, show that either A or B are not invertible.
Show that no 3 x 3 matrix A exists such that A^2 + I = 0.
Any help is appreciated.
You may have another constraint on your matrices you haven't mentioned yet - the matrix (where i=square root of -1):
i,0,0
0,i,0
0,0,i
satisfies the 2nd equation. If you demand that the matrix eigenvalues are real, the argument Tonio mentions should work.