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.

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- Dec 9th 2009, 11:10 AMBrownianManDeterminants & 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. - Dec 9th 2009, 01:52 PMtonio

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

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

Tonio - Dec 9th 2009, 02:00 PMqmech
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.