Matrices can be used in an entirely different manner to solve a system of linear first-order differential equations. Recall that the simple linear first-order differential equation x′ = ax, where a is a constant, has the general solution x = ceat. It seems natural, then, to ask whether we can define a matrix exponential eAt, where A is a matrix of constants, so that eAt is a solution of the system X′ = AX.
We shall now see that it is possible to define a matrix exponential eAt so that the homogeneous system X′ = AX, where A is an n × n matrix of constants, has a solution
X = eAtC.(1) ...