Using the fact that $AC^T = (\det A)I$ where $A$ is some arbitrary matrix and $C$ is the cofactor matrix for $A$, how can I prove that $\det C = (\det A)^{n-1}$
I really don't know how to progress on this. Tried doing some random operations on both equations to see if I can make a link somewhere and: $AC^T = \det A \implies C^T = A^{-1}\det A \implies \det C^T = (\det A)A^{-1}$
$\det C = (\det A)^{n-1} \implies \det C^T = (\det A)^{n-1}$
To be honest, I'm not even sure if the last operation I applied on the first line and the operation I applied on the second line are even legal operations.
Suggestions are welcome.