Abstract We obtain a nontrivial bound on the number of solutions to the equation $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$with a fixed $n\times n$ matrix $A$ over a finite field ${{\mathbb {F}}}_q$ of $q$ elements of multiplicative order $\tau $. We apply our result to obtain a new bound for additive character sums with a matrix exponential function, nontrivial beyond the square-root threshold. For $n=2$, this equation has been considered by Kurlberg and Rudnick (for $\nu =2$) and Bourgain (for large $\nu $) in their study of quantum ergodicity for linear maps over residue rings. We use a new approach to improve their results and also obtain a bound on Kloosterman sums over small subgroups, of size below the square-root threshold.