We now consider the more general matrix case:
where A is n x n , B is n x m, x(t) is n x 1 and u(t) is m x 1. Let us define the matrix f (t) which solves the following matrix differential equation:
where I is the n x n identity matrix. By using Laplace transforms we can transform the above equation into
and thus
and
which is the inverse Laplace transform. One can also verify that f (t) is equal to the following infinite sum by showing that the sum converges and satisfies the above matrix differential equation.
Observe that we can use f (t) as an integrating factor in the following way:
Note that for one of the steps above we have used the fact that f (t) commutes with A; that is Af (t)= f (t)A. Using the infinite sum expression for f (t), the reader can easily verify this fact. In a similar way to the scalar case we arrive at
In the general case where the initial condition is given at some point t0, we have
where
is given. This is the matrix form of the variation of parameters formula. The matrix 
is called the state transition matrix.
Example 1 Solve the following matrix differential equation.
Suppose u(t) = 0.
We first find
Therefore,
and
Hence for the unforced case (u(t) = 0), we have by the variation of parameters formula
Therefore, the solution is given by
Example 2 Solve the following differential equation with a non-zero input.
This time let u(t) = sin2t .
We may use the f (t) which we found previously since no part of its derivation has changed. Hence, we have
Now using the variation of parameters formula given above, we can see that our system satisfies the equation:
Therefore, the solution is given by
Example 3 For the shown spring-mass-damper system, find the position and velocity of the mass if the initial position is 1 m, the initial velocity is zero and the external force f (t ) = 5 N.
The state variables are the position and velocity of the mass. Recall that the state space representation for this system is written as
We need to solve this equation in order to find the state variables. The variation of parameters formula will be used to solve the problem. We first calculate the state transition matrix
The variation of parameters formula gives
