Unimodular matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a unimodular matrix M is a <a href="/facts/Square_matrix/5TP9mNNn">square</a> <a href="/facts/Integer_matrix/qAEpDzIU">integer matrix</a> having <a href="/facts/Determinant/eGYqJ2TF">determinant</a> +1 or −1. Equivalently, it is an integer matrix that is <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> over the <a href="/facts/Integer/PUwwrawx">integers</a>: there is an integer matrix N that is its inverse (these are equivalent under <a href="/facts/Cramer%2527s_rule/aUt4Cena">Cramer's rule</a>). Thus every equation Mx = b, where M and b both have integer components and M is unimodular, has an integer solution. The n × n unimodular matrices form a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> called the n × n <a href="/facts/General_linear_group/B54gNILS">general linear group</a> over 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
, which is denoted 
 
 
 
 
 GL
 
 n
 
 
 ⁡
 (
 
 Z
 
 )
 
 
 {\displaystyle \operatorname {GL} _{n}(\mathbb {Z} )}
 
.

Unimodular matrix open-in-new

Unimodular matrix