Summary: Let \(A\) be an invertible \(d\times d\) matrix with integer elements. Then \(A\) determines a self-map \(T\) of the \(d\)-dimensional torus \(\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d\). Given a real number \(\tau>0\), and a sequence \(\{z_n\}\) of points in \(\mathbb{T}^d\), let \(W_\tau\) be the set of points \(x\in\mathbb{T}^d\) such that \(T^n(x)\in B(z_n,e^{-n\tau})\) for infinitely many \(n\in\mathbb{N}\). The Hausdorff dimension of \(W_\tau\) has previously been studied by Hill–Velani and Li–Liao–Velani–Zorin. We provide complete results on the Hausdorff dimension of \(W_\tau\) for any expanding matrix. For hyperbolic matrices, we compute the dimension of \(W_\tau\) only when \(A\) is a \(2 \times 2\) matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension \(d\).