Unimodular Matrices

A square matrix M of order m is called unimodular if all its entries are integers and its determinant is either +1 or -1: $$ \det(M) = \pm 1 $$

A Practical Example

Consider the following 2×2 matrix:

$$ \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} $$

This matrix is unimodular because its determinant is 1:

$$ \det \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} = 3 \cdot 1 - 2 \cdot 1 = 1 $$

Example 2

Now consider this matrix:

$$ \begin{pmatrix} 1 & 1 \\ 3 & 2 \end{pmatrix} $$

This is also unimodular since its determinant equals −1:

$$ \det \begin{pmatrix} 1 & 1 \\ 3 & 2 \end{pmatrix} = 1 \cdot 2 - 1 \cdot 3 = -1 $$

Unimodular matrices have several important algebraic properties:

The product of two unimodular matrices is also unimodular.
Example. Multiplying two unimodular matrices $$ \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 \\ 3 & 2 \end{pmatrix} = \begin{pmatrix} 3 \cdot 1 + 2 \cdot 3 & 3 \cdot 1 + 2 \cdot 2 \\ 1 \cdot 1 + 1 \cdot 3 & 1 \cdot 1 + 1 \cdot 2 \end{pmatrix} = \begin{pmatrix} 9 & 7 \\ 4 & 3 \end{pmatrix} $$ yields another unimodular matrix, since $$ \det \begin{pmatrix} 9 & 7 \\ 4 & 3 \end{pmatrix} = 9 \cdot 3 - 7 \cdot 4 = 27 - 28 = -1 $$
The inverse of a unimodular matrix is also unimodular.
Example. The unimodular matrix $$ M = \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} $$ is invertible, and its inverse is $$ M^{-1} = \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix} $$ which is itself unimodular, since $$ \det M^{-1} = \det \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix} = 1 \cdot 3 - (-2) \cdot (-1) = 3 - 2 = 1 $$
All identity matrices are unimodular.
Example. The 2×2 identity matrix $$ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ is unimodular because $$ \det I_2 = \det \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = 1 $$

And so on.