(a) Prove using induction that the forward pass of this algorithm always terminates for augmented matrix A ∈ R n×(n+1) . (b) Prove using induction that the backward pass of the algorithm terminates if reached. (c) Prove using induction that the downward pass of the algorithm terminates with an uppertriangular matrix — all the entries below the diagonal are zero. (d) Prove that the algorithm is correct for augmented matrix A ∈ R n×(n+1) for the case when a unique solution exists. To do this, you should first prove a lemma that shows that any solution to the original system of equations remains a solution to the modified system of equations at all iterations of the downward pass of the algorithm, and vice-versa: all solutions of the modified system of equations at all iterations are valid solutions to the original system of equations.