Answer
See below
Work Step by Step
Assume $A$ and $B$ are $n\times n$ matrices with $AB = -BA$ and $n$ is an odd number.
Since $AB = -BA$, then $\det(AB) = \det(-BA)$.
We also know that $\det(AB) \\= \det(A) \det(B)$ and $\det(-BA)\\= \det(-B) \det(A)\\=(-1)^n \det(-B) \det(A) \\=(-1)^n \det(-B) \det(A)$.
Since $n$ is odd, we have$ (-1)" = -1$, and then $\det(-BA) = –\det(A)\det(B)$.
As a result, $\det(A)\det(B)=-\det(A)\det(B)$ which is also $\det(A)\det(B)=0$, resulting in $\det(A)=0$ or $\det(B)=$.
We know that at least one of the matrices $A$ or $B$ is not invertible since a $n \times n$ matrix is invertible if and only if its determinant is nonzero.
