Write out the details of the proof of the "if" part of Theorem 1.7: if A is nonsingular, then the linear system Ax = b has a unique solution for every b.
Theorem 1.7
A linear system Ax = b has a unique solution for every choice of right hand side b if and only if its coefficient matrix A is square and nonsingular.
We are able to prove the "if" part of this theorem, since nonsingularity implies reduction to an equivalent upper triangular form that has the same solutions as the original system. The unique solution to the system is then found by Back Substitution. The "only if" part will be proved in Section 1.8.
The revised version of the Gaussian Elimination algorithm, valid for all nonsingular coefficient matrices, is implemented by the accompanying pseudocode program. The starting point is the augmented matrix M = (A | b) representing the linear system Ax = b. After successful termination of the program, the result is an augmented matrix in upper triangular form M - (U | c) representing the equivalent linear system Ux = c. One then uses Back Substitution to determine the solution x to the linear system.