Does anybody have a solution to the given word problem below?
Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is invertible and and that A-inverse is lower triangular. [HINT: Explain why A can be changed into I using only row replacements and scaling. (Where are the pivots?) Also, explain why the row operations that reduce A to I change I into a lower triangular matrix.]
Big Hint: Think about row reducing [A I]
Best Answer
Do you believe that the product of two lower triangular matrices is again a lower triangular matrix? Row reducing $[A\;I]$ amounts to multiplying $[A\;I]$ by certain matrices which correspond to elementary row operations. The only matrices you need for this particular row reduction are all lower triangular...