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The Bareiss algorithm is fraction-free Gaussian elimination for exact determinants. For an integer matrix, put $A^{(0)}=A$ and $p_0=1$. At step $k$, choose a nonzero pivot element $p_k=a_{kk}^{(k-1)}$ and update the trailing entries bySet the eliminated entries below the pivot to zero. The determinant identities behind the method make division by $p_{k-1}$ exact, so intermediate entries remain integers. Without row exchanges, the last pivot $a_{nn}^{(n-1)}$ is $\det A$; row exchanges change its sign in the usual way. A zero pivot needs an exchange before continuing.

For $A=\left(\begin{array}{ccc}2& 1& 0\\ 1& 2& 1\\ 0& 1& 2\end{array}\right)$, the first trailing block becomes $\left(\begin{array}{cc}3& 2\\ 2& 4\end{array}\right)$. The last update is $(3\cdot 4-2\cdot 2)/2=4$, giving the determinant without rational intermediate entries. Exact division is different from having no division at all.^[ref]

Erwin H. Bareiss developed integer-preserving methods for Gaussian elimination. The Bareiss algorithm uses determinant identities to arrange divisions that are exact, reducing the growth of intermediate rational expressions.^[ref]

An update combines two products and divides by a previous pivot. The reason this stays in the integers is structural: the quantities represent related minors satisfying Sylvester identity for bordered minors. This makes exact linear algebra practical without replacing the calculation by floating-point approximation.^[ref]

Let a square matrix $A$ have a leading $k\times k$ block $B$. For row and column indices $i,j>k$, let $D_{ij}$ be the determinant formed by bordering $B$ with row $i$ and column $j$, keeping that order. For $i<r$ and $j<s$, the Sylvester identity for bordered minors isIt is a polynomial identity and remains valid when $B$ is singular; dividing by $\det B$ requires an additional nonzero condition.^[ref]

For $k=1$ and $A=\mathrm{diag}(2,3,5)$, the bordered minors are $D_{22}=6$, $D_{33}=10$, $D_{23}=D_{32}=0$. The identity reads $60=2\cdot 30$. In Bareiss algorithm, this identity explains why division by a previous pivot can be exact over the integers.