# The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.

Put content here**Theorem.** If $A$ is an invertible upper triangular matrix, then $A^{-1}$ is also upper triangular. Similarly, if $A$ is an invertible lower triangular matrix, then $A^{-1}$ is lower triangular.
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**Note:** A triangular matrix is invertible if and only if all its diagonal entries are nonzero.
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**Proof idea.** For an upper triangular matrix $A$, the inverse can be computed by back-substitution, which preserves the upper triangular structure.
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**Example.** The inverse of $A = \begin{pmatrix} 2 & 3 \\ 0 & 4 \end{pmatrix}$ is $A^{-1} = \begin{pmatrix} 1/2 & -3/8 \\ 0 & 1/4 \end{pmatrix}$, which is also upper triangular.
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The diagonal entries of $A^{-1}$ are the reciprocals of the diagonal entries of $A$: $(A^{-1})_{ii} = 1/a_{ii}$.

# Parents

* Triangular matrices