# 1. Show that if A is an upper triangular matrix and invertible then A^-1 is also upper triangular?

2. Show that if A is an lower triangular matrix and invertible then A^-1 is also lower triangular

### 1 Answer

Relevance

- Simon van DijkLv 66 years agoFavorite Answer
Picture the product of A with its inverse. A is a upper triangulaer matrix (of size n) and B (the inverse) is a as yet matrix with elements B(i,j). Their product is the identity matrix.

Focus on the last minus -1 element of the last row of the identity matrix: its value 0. It is created by the sum of A(n,j)*B(j,n-1) = 0

As all elements of the last row of A are 0 (except for the last), this sum = A(n,n).B(n,n-1) = 0 As A(n,n_ not equal to zero, so B(n-1,n) must be. So you can work out from B(n-1,n) until B(2,1), that all the elements in the lower part must be zero.

The same line of reasoning applies for the lower triangular matrix.

Still have questions? Get your answers by asking now.