dx / dt = 1x  2y equation (1)
dy / dt = 4x + 4y equation (2)
In matrix form we can write these equations as:
dx / dt 
= 
1 
2 
* 
X 
dy / dt 

4 
4 

Y 
1 
2 
4 
4 
X 
Y 
Where, I =
1 
0 
0 
1 
1 – L 
 2 
= 
0 
4 
4 – L 


(1 – L) (4 – L) – (4) (2) = 0,
Or 4 – L + L2 – 4L – ( 8) = 0,
OR L2 – 5L + 4 = 0,
OR L = 4 and L = 1,
Next we calculate eigen vectors on basis of values of 'L' using the formula: A P = L P. Where, 'P' is a column matrix of eigen vectors. Diagonalize the matrix 'A' by performing following matrix operation: D = P A (inverse of P).
Where, 'D' is a matrix that only has values on its diagonal. Rewrite the matrix 1 by substituting X = PY and D = PA (inverse of P) to get:
dY / dt = DY equation 1
Last step is to find out integral of every row of matrix equation 1 to find solutions for 'Y'. Substitute the value of 'Y' in equation X = PY to get solutions for original equation. This is how you solve system of coupled differential equations using matrix representation.