This paper studies many homological properties of modules over a formal triangular matrix ring \(T=\left(\begin{smallmatrix} A&0\\ M&B\end{smallmatrix}\right)\), where \(A\) and \(B\) are rings and \(M\) is a \(B\)-\(A\)-bimodule. Up to equivalence of categories, a right \(T\)-module \(K\) is a triple \((X,Y)_f\), where \(X\) is a right \(A\)-module, \(Y\) is a right \(B\)-module, and \(f\colon(Y\times M)\to X\) is a suitable map which, in effect, gives a way of multiplying elements of \(Y\) and elements of \(M\) to produce elements of \(X\). One can then think of \(K\) as being the row-vector \((X,Y)\), and the right action of \(T\) on \(K\) is given by matrix multiplication. Necessary and sufficient conditions for \(K\) to be injective are already known, and these are extended here to determine when \(K\) has injective dimension at most \(n\). Similarly it is determined when \(K\) has projective dimension at most \(n\), and when \(K\) satisfies other conditions such as being cotorsion, tilting, etc.