Illustrate the general system of n first-order linear equations?

What is System of first order Linear Equation:

When there is more than one first order differential equations containing one independent variable and more than dependent variable, the system is referred to as a system of first order linear differential equation. Every higher order linear differential equation can be converted to a system of first order linear differential equations. The order of the parent equation determines the number of equations in the system. 

To Discuss:

We discuss the general system of n first-order linear equations.

A system of $n$ first order linear differential equation is represented by the equation,

$x\text{'}=Ax+b\left(t\right) \left(1\right)$

Here, $A\in M_{n\times n}\left(F\right)$ and $x\in {\mathrm{ℝ}}^n$.

When $b\left(t\right)=0$, the system is referred to as a homogeneous system. When $b\left(t\right)\ne 0$, the system is non-homogeneous. 

For a homogeneous system, the general solution is only the complementary function which is the solution to the equation

$x\text{'}=Ax$

The solution to the homogeneous system is

$x\left(t\right)=\sum _{i=1}^n{c}_i{v}_i{e}^{{\lambda }_{i}t}$

Here, $c_i, i=1,2,3,...,n$ are arbitrary constants, ${\lambda }_i, i=1,2,3,...,n$ are the eigenvalues corresponding to the matrix $A$ and $v_i, i=1,2,3,..,n$ are the respective eigenvectors.