In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once.
If the set of vertices is then the Tutte matrix is an n-by-n skew-symmetric matrix A with entries
where the x<sub>ij</sub> are indeterminates. The determinant of this matrix is then a polynomial (in the variables x<sub>ij</sub>, i < j ): this coincides with the square of the Pfaffian of the matrix A and is non-zero (as a polynomial) if and only if a perfect matching exists. (This polynomial is not the Tutte polynomial of G.)
The Tutte matrix is named after W. T. Tutte, and is a generalisation of the Edmonds matrix for a balanced bipartite graph.