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I already have the result that says that if $U$ is upper triangular and non singular then $U^{-1}$ is also upper triangular. I want to use this result to prove the result for lower triangular matrix $n \times n$.

TRY:

Let $A$ be a lower triangular matrix which is invertible. Let $U = A^T$. Then $U$ is upper triangular and invertible. hence $U^{-1}$ is upper triangular as well. In other words $(A^T)^{-1} = (A^{-1})^T $ is upper triangular. Taking the transpose of the transpose give us then that $A^{-1}$ must be lower triangular. Is this a valid proof ?

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    $\begingroup$ This proof is valid if you're allowed to use the assumption about upper-triangular matrices. $\endgroup$ – Omnomnomnom Sep 7 '15 at 23:26

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