I am asked the following:
Assume that $M$ is skew-symmetric and invertible. If $M$ is an $n \times n$ matrix, prove that $n$ is even.
The following is my attempt at the solution. I'm not too sure if it is right, or if I'm allowed to do this.
So, if a matrix is skew-symmetric, then,
and by taking the determinant of both sides, $$\det(A^T)=\det(-A)$$
Using the property of transpose and the multi-linear property,
Here is where my problem arises. Are we allowed to assume that $n$ is even and then show that it is invertible? However, if we do this we get
$$\det(A) = \det(A)$$
and I'm not too sure what this equality means. Or where to go from here. Any help would be greatly appreciated.