Let $A$, $B$, and $C$ be three $n\times n$ matrices. We investigate the problem of verifying whether $AB=C$ over the ring of integers and finding the correct product $AB$. Given that $C$ is different from $AB$ by at most $k$ entries, we propose an algorithm that uses $O(\sqrt{k}n^2+k^2n)$ operations. Let $\alpha$ be the largest integers in $A$, $B$, and $C$. The largest value involved in the computation is of $O(n^3\alpha^2)$.

翻译：设 $A$, $B$, $C$ 为三个 $n\times n$ 矩阵。我们研究在整数环上验证 $AB=C$ 并找出正确乘积 $AB$ 的问题。假设 $C$ 与 $AB$ 至多有 $k$ 个不同元素，我们提出一种使用 $O(\sqrt{k}n^2+k^2n)$ 次运算的算法。设 $\alpha$ 为 $A$, $B$, $C$ 中的最大整数，计算过程中最大数值量为 $O(n^3\alpha^2)$。