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.

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:

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.

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