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A square matrix is invertible iff its determinant is invertible. Consider the QR-decomposition of an $n \times n$ matrix $A$, i.e., $A = QR$ where $Q$ is upper-triangular and $R$ consists of $n$ orthogonal columns such that $Q^\mathsf{T}Q = I_n$. Then we can compute the determinate of $A$ quickly by just multiplying the diagonal elements of $Q$. Moreover, ...
Assume you know the number of invertible $m \times m$ matrices over $\mathbb{Z}_{p^k}$ for $p$ prime. Call this number $N(p^k)$. By the Chinese Remainder Theorem, $N(\prod_{i=1}^l p_i^{k_i}) = \prod_{i=1}^l N(p_i^{k_i})$ because it's invertible if every component in the CRT decomposition is invertible. For $m = 1$, it's just the $\phi$ function. Finding ...