There is a relation between fractional-order derivatives and a first-order matrix operator raised to a fractional power. We will restrict ourselves here and consider only one part of the process, viz. the evaluation of a first-order matrix operator raised to a fractional exponent. How exactly this is done numerically using MATLAB is described here. In the process, we will define the bandwidth of a matrix and watch its growth concerning the exponent and rank of the matrix. We will also find the difference between a derivative with an integer exponent and a fractional one. A comparison of the fractional derivative of the exponential function is also conducted to show how precise the equivalence of the formulation of a backward-in-time matrix operator raised to a certain fractional power is.