How to prove inverse of Matrix If you have an invertible upper triangular matrix $M$, how can you prove that $M^{-1}$ is also an upper triangular matrix? I already tried many things but don't know how to prove this. Please help!

Look at the algorithm of invertion. You take yor matrix $A$ and matrix $E$, reduce $A$ to $E$ and perform same conversions to $E$. Then you have $E$ and $A^{-1}$. Note just that $E$ is upper triangular and any reducing transformations do not affect its triangularity.
