# Thread: Inverse of upper triangular matrix with diagonal elements all 1

1. ## Inverse of upper triangular matrix with diagonal elements all 1

Prove that the inverse of an upper triangular matrix whose diagonal entires are all 1 is itself an upper triangular matrix whose diagonal entires are all 1.

I know that we can use Cramer's rule and just crunch out the result but I was hoping that someone could provide a short and simple reason.

2. Using matrix multiplication, it is (fairly) easy to show that upper triangular matrices are closed under multiplication. If you haven't seen this before, let A,B be upper triangular and C=AB. Then letting
$\displaystyle A=[a_{ij}], B=[b_{ij}], C=[c_{ij}]$, the rules of matrix multiplication show $\displaystyle c_{ij}=\displaystyle\sum_{k=1}^n a_{ik}b_{kj}$. But for $\displaystyle k<i$ we have $\displaystyle a_{ik}=0$ since upper triangular. Similarly for $\displaystyle k>j$ we have $\displaystyle b_{kj}=0$. But if $\displaystyle i>j$ then the previous arguments imply $\displaystyle a_{ij}=0$.
But these are just the strictly lower triangular entries.

The same argument yields that $\displaystyle c_{ii}=a_{ii}b_{ii}$.