| # Solving triangular systems of equations: backwards substitution example Consider the triangular system we solve for the last variable first, obtaining (from the last equation) . We plug this value of into the first and second equation, obtaining a new triangular system in two variables : We proceed by solving for the last variable . The last equation yields . Plugging this value into the first equation gives . We can apply the idea to find the inverse of the square upper triangular matrix , by solving The matrix is then the inverse of . We find As illustrated above, the inverse of a triangular matrix is triangular.