# Thread:

1. ## Determinants & Matrix Inverses

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.

2. Originally Posted by BrownianMan
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.

Since $\det(AB)=\det(A)\det(B)$ , we get $AB=-BA\Longrightarrow \det(AB)=\det(-BA)=\det(-B)\det(A)=(-1)^n\det(B)\det(A)$.

Now suppose both matrices are invertible (i.e., their determinant is non-zero), and get a huge contradiction.

Tonio

3. 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.
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