### 10.5 Matrix Exponential

## INTRODUCTION

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* = *ce*^{at}. It seems natural, then, to ask whether we can define a matrix exponential *e*^{At}, where **A** is a matrix of constants, so that *e*^{At} is a solution of the system **X**′ = **AX**.

### Homogeneous Systems

We shall now see that it is possible to define a **matrix exponential** *e*^{At} so that the homogeneous system **X′** = **AX**, where **A** is an *n* × *n* matrix of constants, has a solution

**X** = *e*^{At}**C**.(1) ...

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