We can use the computational method described in this section in hopes of finding a matrix inverse, as Theorem CINM gets us halfway there. We will continue with the matrix from Example MI. First we check that the matrix is nonsingular so we can apply the theorem, then we get “half” an inverse, and verify that it also behaves as a “full” inverse by meeting the full definition of a matrix inverse (Definition MI).
Note that the matrix J is constructed by taking the last 5 columns of N (numbered 5 through 9) and using them in the matrix_from_columns() matrix method. What happens if you apply the procedure above to a singular matrix? That would be an instructive experiment to conduct.
With an inverse of a coefficient matrix in hand, we can easily solve systems of equations, in the style of Example SABMI. We will recycle the matrices A and its inverse, J, from above, so be sure to run those compute cells first if you are playing along. We consider a system with A as a coefficient matrix and solve a linear system twice, once the old way and once the new way. Recall that with a nonsingular coefficient matrix, the solution will be unique for any choice of const, so you can experiment by changing the vector of constants and re-executing the code.