A rational matrix is a matrix whose entries are all rational numbers. The set
of 
rational matrices is denoted 
and is a vector space
over the field 
. After bases are chosen, its elements represent the linear
transformations from 
to 
.
Rational matrices are closed under matrix addition, multiplication by rational scalars, and matrix multiplication when the dimensions are compatible. Every integer matrix is rational, and every rational matrix is both a real matrix and a complex matrix. Conversely, multiplying a rational matrix by a common denominator of its entries gives an integer matrix.
The Hilbert matrix is a rational matrix whose matrix inverse has entries in the integers. The Pascal matrix and Redheffer matrix are integer, and therefore rational, matrices.
A rational square matrix is invertible over 
iff its determinant
is nonzero. In this case, its matrix inverse is
rational by the adjugate matrix formula. Its characteristic polynomial has rational
coefficients, although its eigenvalues
need not be rational. For example, the rational matrix
![A=[0 2; 1 0]](/images/equations/RationalMatrix/NumberedEquation1.svg) |
has eigenvalues 
and 
.
