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Example 8.12
Using the Cayley-Hamilton technique, solve the system of equations:

dx
1 = x + 2 x
dt 1 2
dx
2 = 2 x − 2 x
dt 1 2

with the initial conditions: x (0) = 1 and x (0) = 3
1
2
Solution: The matrix A for this system is given by:

 1 2 
A =  
2 −  2

and the solution of this system is given by:

At
X(t) = e X(0)

Given that A is a 2 ⊗ 2 matrix, we know from the Cayley-Hamilton result
that the exponential function of A can be written as a first-order polynomial
in A; thus:

P(A) = e = aI + bA
At
To determine a and b, we note that the polynomial equation holds as well for
the eigenvalues of A, which are equal to –3 and 2; therefore:

e −3 t =− b
a 3
t 2
e =+ 2 b
a
giving:

a = 2 e −3 t + 3 e t 2
5 5
1 1
b = e − e −3 t
t 2
5 5
and

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