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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.

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    $\begingroup$ If the book really proves a theorem, and then gives a counterexample to it, then I would advise you to throw the book. $\endgroup$ Commented Nov 7, 2021 at 16:27
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    $\begingroup$ Probably a typo. The matrix you put in the question is invertible. In the version of Anton and Rorres I just looked at (11th ed), there is almost this example, but with a $0$ (not a $2$) in the row 2, column 2 entry. $\endgroup$ Commented Nov 7, 2021 at 16:31
  • $\begingroup$ hahah, i dont know if i miss something or what. but this book has a good reputation online, and it indeed explain things well $\endgroup$ Commented Nov 7, 2021 at 16:31
  • $\begingroup$ thank you so much, then this must be an error in the book! $\endgroup$ Commented Nov 7, 2021 at 16:32
  • $\begingroup$ Indeed, if it is a typo and the real matrix is as @leslietownes says, then the book is correct. Except for the fact that the matrix is no longer a counterexample to the theorem, but just an example to it. $\endgroup$ Commented Nov 7, 2021 at 16:36

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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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  • $\begingroup$ thank you my friend, i was really thinking whats going on with this one. You guys are awsome, this is my first time post a question, didnt expect reply so fast $\endgroup$ Commented Nov 7, 2021 at 16:41

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