Summary: The group \(G=\mathrm{GL}_r(k)\times (k^\times)^n\) acts on \(\mathbb{A}^{r\times n}\), the space of \(r\)-by-\(n\) matrices: \(\mathrm{GL}_r(k)\) acts by row operations and \((k^\times)^n\) scales columns. A matrix orbit closure is the Zariski closure of a point orbit for this action. We prove that the class of such an orbit closure in \(G\)-equivariant \(K\)-theory of \(\mathbb{A}^{r\times n}\) is determined by the matroid of a generic point. We present two formulas for this class. The key to the proof is to show that matrix orbit closures have rational singularities.