On the definition of totally unimodular matrix

Question

I'm a bit confused about the definition of a totally unimodular matrix, since my lecture notes states that this matrix is not totally unimodular:

$$\begin{pmatrix} 1 && 0 \\ 1 && -1 \\ -2 && 1 \end{pmatrix}$$

I don't see a submatrix which doesn't have a determinant in $\lbrace 0,\pm1 \rbrace$. What am I doing wrong?

Answer 1

From Schrijver's Theory of Linear and Integer Programming (1998):

In Chapters 19 to 21 we consider totally unimodular matrices, which are matrices with all subdeterminants equal to $1$, $-1$, or $0$ (in particular, each entry is $1$, $-1$, or $0$). Totally unimodular matrices yield a prime class of linear programming problems with integral optimum solutions.

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Note that $-2 \notin \{-1,0,1\}$.