Question

Linear Algebra.
Let A and B be 5x5 matrices, such that B is invertible. Suppose that AB =-BA.Show that A is not invertible.

Answer 1

look at the determinant

Answer 2

\[
A B = - B A\\
Det( AB) = (-1)^5 Det(B A)\\
Det(A) Det(B) = -Det(B)Det(A)\\
Det(A) = - Det(A)\\
\]
Since
\[
Det(B) \ne 0
\]
Can you conclude now that A is not invertible?

Answer 3

The above is true for 5 replaced by any odd integer.