Inverse by row operations

You will write the three row operations and use them to find an inverse by turning $[A \,|\, I]$ into $[I \,|\, A^{-1}]$. The program uses exact fractions. A matrix is a list of rows. The checker tests many matrices.

The idea. Swap, scale and add-a-multiple change the rows of the augmented matrix. For each column, find a row with a non-zero entry, move it to the pivot place, scale it to $1$, and clear the rest of the column. If no row has a non-zero entry, the matrix is singular.

Your tasks.

1. Finish add_multiple(M, i, j, k). Change row i to row i plus k times row j.
2. Finish scale(M, i, k). Multiply row i by k.
3. Finish invert(A). Use the operations to return the inverse, or None if A is singular. The swap and the setting up of [A | I] are written for you.