Inverse of an invertible triangular matrix (either upper or lower) is triangular of the same kind How can we prove that the inverse of an upper (lower) triangular matrix is upper (lower) triangular?

The proof is based on cayley–hamilton theorem.
Suppose $A$ is upper triangular matrix. Then we know the characteristic polynomial of $A$ is $P_A(\lambda)=(\lambda-\lambda_1)(\lambda-\lambda_2)...(\lambda-\lambda_n)$. By cayley–hamilton theorem we know $P_A(A)=0$, i.e.$(A-\lambda _1I)(A-\lambda _2I)...(A-\lambda _nI)=0$, from where we can see that $I=A\frac{\sum_i{c_iA^i}}{\underset{i}{\Pi}\left( -\lambda _i \right)}$ where $c_i$ can be solved from $P_A(A)=0$. Compare $I=A\frac{\sum_i{c_iA^i}}{\underset{i}{\Pi}\left( -\lambda _i \right)}$ with $I=AA^{-1}$ we can see that $A^{-1}$ is also upper triangular matrix.
