If you manipulated a matrix so all the entries above the diagonal entries are 0s, how would the solution differ from when made the entries below the diagonal entries 0s?

(Exact Answer) When all the entries above the diagonal entries are 0, the solution will not be different from when all the entries below the diagonal entries are 0. This is because in both cases, there will be one row that would be left with just one unknown element, thus giving us the solution to the other unknowns through back-substitution. For instance, in a linear system with three unknowns, x, y, and z, the traditional triangulated form will give us the value of z first; however, if the entries above the diagonal entries in the matrix are 0, then we will get the value of x first. In either case, the solution will be the same.