- Integer matrix
In

mathematics , an**integer matrix**is a matrix whose entries are allinteger s. Examples include thebinary matrix ; thezero matrix ; theunit matrix ; theadjacency matrix used ingraph theory , amongst many others. Integer matrices find frequent application incombinatorics .For example:

:$left(egin\{array\}\{cccc\}\; 5\; 2\; 6\; 0\backslash \; 4\; 7\; 3\; 8\backslash \; 5\; 9\; 0\; 4\backslash \; 3\; 1\; 0\; 3\backslash \; 9\; 0\; 2\; 1end\{array\}\; ight)$ and $left(egin\{array\}\{ccc\}\; 1\; 5\; 0\backslash \; 0\; 9\; 2\backslash \; 1\; 7\; 3end\{array\}\; ight)$Are both examples of integer matrices.

**Notes**Invertibility of integer matrices is in general more numerically stable than that of non-integer matrices. The

determinant of an integer matrix is itself an integer, thus the smallest possible magnitude of the determinant of an invertible integer matrix is**one**, hence where inverses exist they do not become excessively large (seecondition number ). Theorems frommatrix theory that infer properties from determinants thus avoid the traps induced by ill conditioned ("nearly" zero determinant) real orfloating point valued matrices.The inverse of an integer matrix $M$ is again an integer matrix if and only if the determinant of $M$ is exactly $1$ or $-1$. Integer matrices of determinant $pm\; 1$ form the group $mathrm\{GL\}(n,mathbf\{Z\})$, which has far-reaching applications in arithmetic and geometry. For $n=2$, it is closely related to the

modular group .The

characteristic polynomial of an integer matrix has integer coefficients. Since the eigenvalues of a matrix are the roots of the polynomial, the eigenvalues of an integer matrix arealgebraic integers . In dimension less than 5, they can thus be expressed by radicals involving integers.Integer matrices are sometimes called "integral matrices", although this use is discouraged.

**External links*** [

*http://mathworld.wolfram.com/IntegerMatrix.html Integer Matrix at MathWorld*]

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