Inverse of upper triangular matrix

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Homework Statement

Show that if A is an invertible upper triangular matrix, then A^-1 is also an upper triangular matrix

Homework Equations

The Attempt at a Solution

The inverse of something just flips the main diagonal and leaves everything else where is was just changing the signs. How do I say this in math language.

 

[Mark44]

Homework Statement

Show that if A is an invertible upper triangular matrix, then A^-1 is also an upper triangular matrix

Homework Equations

The Attempt at a Solution

The inverse of something just flips the main diagonal and leaves everything else where is was just changing the signs. How do I say this in math language.

Since it's not true, it's not worth saying in math language.

For example,
$$A = \begin{bmatrix} 1&1\\1&-1 \end{bmatrix}$$

It turns out that this matrix is invertible, and its inverse is
$$A^{-1} = \begin{bmatrix} 1/2&1/2\\1/2&-1/2 \end{bmatrix}$$

So here is "something" (A) that has an inverse, but I don't see that the main diagonal got flipped and only the signs on the off diagonal got changed. Admittedly, this isn't an upper triangular matrix.