In the example on solving systems of linear equations, we swapped the positions of rows 2 and 3. This is known as a permutation.
When we are doing triangular factorization, we want our pivot values to be along the diagonal of the matrix, but this won't happen every time—in fact, it usually won't. So, instead, what we do is swap the rows so that we get our pivot values where we want them.
But that is not their only use case. We can also use them to scale individual rows by a scalar value or add rows to or subtract rows from other rows.
Let's start with some of the more basic permutation matrices that we obtain by swapping the rows of the identity matrix. In general, we have n! possible permutation matrices that can be formed ...
