Proving Triangular Matrix Inverse is Also Triangular

[crd]

It is stated in almost every linear algebra text i could find that the inverse of a triangular matrix is also triangular, but no proofs accompanied such statements.

I am convinced that it is the truth, but I have not been able to write anything down that I am satisfied with that doesn't rely on the argument that row operations on the matrix (A|I) to obtain (I|A^{-1}).

Since this would only be the forward pass(if A is lower triangular) and the backwards pass(if A is upper triangular) and these operations ultimately do not introduce non zero terms above/below the diagonal entries(depending on what A was), thus A^{-1} would be a triangular matrix of the same flavor.

Has anyone come across anything a little more elegant than simply brute forcing it?

 

[trambolin]

Assume you have a invertible upper triangular matrix. Consider

$$AA^{-1} = I$$

You can use induction, starting from the last row of A times the last column of $A^{-1}$. gives you the entry. lower left entry 1.

Then again take the last row of A and n-1 column of $A^{-1}$. To be able to get a zero in the identity matrix, (n,n-1) entry of $A^{-1}$ must be zero.
...
Carry on to the upper left corner and you are done.