Inmathematics, aninteger matrix is amatrix whose entries are allintegers. Examples includebinary matrices, thezero matrix, thematrix of ones, theidentity matrix, and theadjacency matrices used ingraph theory, amongst many others. Integer matrices find frequent application incombinatorics.

    and    

are both examples of integer matrices.

Invertibility of integer matrices is in general more numerically stable than that of non-integer matrices. Thedeterminant of an integer matrix${\displaystyle M}$ is itself an integer, and theadjugate matrix of an integer matrix is also integer matrix. Thus, the determinant of an invertible integer matrix is${\displaystyle 1}$ or${\displaystyle -1}$, and 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 byill conditioned (nearly zero determinant)real orfloating point valued matrices.

The inverse of an integer matrix${\displaystyle M}$ is again an integer matrix if and only if the determinant of${\displaystyle M}$ equals${\displaystyle 1}$ or${\displaystyle -1}$.^[1] Integer matrices of determinant${\displaystyle 1}$ form thegroup${\displaystyle {\mathrm{S}\mathrm{L}}_n(\mathbf{Z})}$, which has far-reaching applications in arithmetic andgeometry. For${\displaystyle n=2}$, it is closely related to themodular group.

The intersection of the integer matrices with theorthogonal group is the group ofsigned permutation matrices.

Thecharacteristic polynomial of an integer matrix has integer coefficients. Since theeigenvalues of a matrix are theroots of this polynomial, the eigenvalues of an integer matrix arealgebraic integers. In dimensionless than 5, they can thus be expressed byradicals involving integers.

Integer matrices are sometimes calledintegral matrices, although this use is discouraged.

• GCD matrix
• Unimodular matrix
• Wilson matrix

• Integer Matrix at MathWorld

• Matrices (mathematics)