How to prove $\det{AB} = \det{BA} = \det{A}\det{B}$?

Question

Possible Duplicate:
How to show $\det(AB) =\det(A)\det(B)$

How would I prove that

$$\det{AB} = \det{BA} = \det{A}\det{B}$$

Answer 1

From Wikipedia: This property is a consequence of the characterization [given above] of the determinant as the unique $n$-linear alternating function of the columns with value $1$ on the identity matrix, since the function $M_n(K)\to K$ that maps $M\mapsto\det(AM)$ can easily be seen to be $n$-linear and alternating in the columns of $M$, and takes the value $\det(A)$ at the identity.