The authors start by proving some interesting results in low-dimensional matrix approximation problems. For example, they show that (1) a one-dimensional subspace \({\mathcal L}\) of \(M_n(\mathbb{C})\), the space of \(n\)-by-\(n\) complex matrices, is Chebyshev (that is, every matrix in \(M_n(\mathbb{C})\) has a unique approximant in \({\mathcal L}\)) if and only if the nonzero matrix in \({\mathcal L}\) is invertible, and (2) a two-dimensional subspace of \(M_2(\mathbb{C})\) is Chebyshev if and only if it contains two linearly independent rank-one matrices. Furthermore, let \({\mathcal D}\) denote the subalgebra of diagonal matrices in \(M_3(\mathbb{C})\). It is shown that if \(A\) in \(M_3(\mathbb{C})\) is such that \(d_{\mathcal D}(A)> \delta_{\mathcal D}(A)\), where \(d_{\mathcal D}(A)= \inf\{\| A-B\|:B\) in \({\mathcal D}\}\) and \(\delta_{\mathcal D}(A)= \sup\{\|(I- P_M)AP_M\|: M\) invariant subspace for all matrices in \({\mathcal D}\}\), \(P_M\) being the orthogonal projection onto \(M\), then \(A\) has a unique approximant in \({\mathcal D}\). The constants \(d_{\mathcal L}(A)\) and \(\delta_{\mathcal L}(A)\) for a subalgebra \({\mathcal L}\) of \(M_n(\mathbb{C})\) are the ones used in defining the notion of hyperreflexivity (\({\mathcal L}\) is hyperreflexive if there is \(C> 0\) such that \(d_{\mathcal L}(A)\leq C\delta_{\mathcal L}(A)\) for all \(A\) in \(M_n(\mathbb{C})\)), which was studied intensively in recent years. To study the well-posedness of the approximation process, the authors define the notion of an almost Chebyshev subspace: a subspace \({\mathcal L}\) of \(M_n(\mathbb{C})\) is almost Chebyshev if there is an open dense subset of \(M_N(\mathbb{C})\) consisting of matrices \(A\) which have a unique approximant in \({\mathcal L}\). The paper concludes with the examination of the above three subspaces for being almost Chebyshev.