Summary: It is shown that if \(f(z)=\sum^ \infty_{i=0} a_ i z^ i\) (the series having radius of convergence \(R\)) then for any \(n\times n\) matrix \(A\) with spectral radius less than \(R\) and any norm \(\|\cdot\|\) on the space of \(n\times n\) matrices \[ \left\| f(A)-\sum^ k_{i=0} a_ i A^ i\right\|\leq {1\over (k+1)!} \max_{s\in [0,1]}\| A^{k+1} f^{(k+1)} (sA)\|. \] It is also shown that expressions for the error in numerical integration rules can be generalized to matrix- valued functions. The main point of the paper is that in these two cases it is not necessary to increase the bound by a factor depending on \(n\) when generalizing an inequality for scalar-valued functions to \(n\times n\) matrix-valued functions.