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.

## Best Answer

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:

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.