Are there any good tricks for finding the inverse of a matrix via Gauss-Jordan elimination when that matrix has lots of zeroes?

Question

Suppose we're given a square matrix $A$, say $4 \times 4$. If we're asked to find it's determinant by hand, a nice trick is to find a row or column with lots of zeroes (examiners love giving these kinds of matrices, who really knows why...) along which to perform a cofactor expansion.

Suppose that instead of being asked to find the determinant, we're asked to find the inverse by Gauss-Jordan elimination. In this case, it's less clear to me how to exploit the rows and columns with many zeroes to simplify the problem.

Question. Are there any good tricks for finding the inverse of matrix with many zeroes via Gauss-Jordan elimination?

For example, to invert something like this:

$$\begin{bmatrix} 1 & 0 & 2 & 3 \\ 2 & -1 & 1 & 0 \\ -1 & 0 & 2 & 1 \\ 3 & 0 & 3 & 2 \end{bmatrix}$$

are there any nice tricks for exploiting that column of mostly $0$'s?

Answer 1

Interchanging the first two columns of $A$ corresponds to interchanging the first two rows of $A^{-1}$, and interchanging the first two rows of $A$ corresponds to interchanging the first two columns of $A^{-1}$. After doing those interchanges, you have a matrix where the first column of Gauss-Jordan has been (almost) done already.