TITLE: Does swapping columns of a matrix cause the rows of the inverse matrix to be swapped?
QUESTION [16 upvotes]: This question came up while I was performing some computation on a few matrices on an unrelated problem in computer science.
Let $A$ be an invertible matrix with columns $A_1, \dots A_n$. Let $B$ be its inverse, with rows $B_1, \dots, B_n$. Now construct a new matrix $\hat{A}$ by swapping two columns $A_i$ and $A_j$. Let $\hat{B}$ be the inverse of $\hat{A}$.
In my specific computations, I noticed that $\hat{B}$ was nothing more than $B$ with the rows $B_i$ and $B_j$ swapped. Is this always true? I doubt it was a coincidence because the numbers were quite random.
Note: I was working with real numbers but I'd be interested to know if the field makes any difference.

REPLY [13 votes]: Another way to think of it: you are just re-numbering the basis for one of the vector spaces:  The domain space for $A$, which is the range space fo $A^{-1}$.