# Triangular matrix is invertible counterexample

linear algebra

when i was reading the book "elementary linear algebra with applications" by Howard Anton, Chris Rorres. There is a theorem said

A triangular matrix is invertible iff its diagonal entries are all non zero.

I know how to proof this theorem, but he immediately shows a counterexample without explaining it.
the counter example is as follows:
$$\begin{bmatrix}3&-2&2\\0&2&-1\\0&0&1\end{bmatrix}$$

My question is why this is not invertible, and why this theorem does not hold for this case. Further, is there a strong statement to conclude this theorem. Many thanks.

This is a typo in the $$9^{\text{th}}$$ edition (and maybe earlier editions) of the textbook. The $$10^{\text{th}}$$ edition and later editions correctly say:
Consider the upper triangular matrices $$A = \begin{bmatrix}1&3&-1\\0&2&4\\0&0&5\end{bmatrix}\qquad B = \begin{bmatrix}3&-2&2\\0&0&-1\\0&0&1\end{bmatrix}$$It follows from part (c) of Theorem 1.7.1. that the matrix $$A$$ is invertible, since its diagonal entries are nonzero, but the matrix $$B$$ is not.
Note the $$0$$ in the center of matrix $$B$$, corrected from the $$2$$ in your edition. The text in italics is also added in the $$10^{\text{th}}$$ edition.