Interchanging two columns of a matrix will lead to change in the sign of determinant Interchanging two columns of a square matrix changes the sign of the determinant.
I know that this is true, and I do understand how it works. But is there any proof for this statement?

*

*Determinant of the matrix 
$$J = \begin{bmatrix}0&1\\1&0\end{bmatrix}$$ is $-1$. 

*Determinant of block diagonal matrix $\det{\textrm{diag}(A,B)}$ is $(\det A)\times(\det B)$

*determinant of a product of two matrices $AB$ is again $(\det A)\times(\det B)$
Now a row swap is equivalent to 
$$
\begin{bmatrix}
I_k & & & \\ 
  &0&1& \\
  &1&0& \\
  & & &I_l
\end{bmatrix}A = PA
$$
The determinant of $P$ is first by using the second property $\det(I_k)\det(J)\det(I_l)$. Now by using the first identity we have $\det(P) = 1\times-1\times1 = -1$. 
Now using the third property $\det(PA) = -\det(A)$.
If you work from the right side of $A$ this corresponds to a column swap.
