so now we have where and where

and the matrix is . Its determinant is (e^-t)(-2e^-4t)-(e^-t)(e^-4t)= - 3e^-5t

Its inverse^-1 is

Note: The matrix can be written as the product of the T and Q matrices ONLY if the roots of the eigenvalues equation are distinct

Suppose you wanted to solve +

Since we have already solved for the matrix and found the inverse, namely, ^-1 is , we have to find

=

integrate this to get d = and this must be multiplied by to get ζ_p

that is ζ_p= X = and, finally,

zp=

The total solution is then . It should be noted that it is on this that we apply the initial conditions.

You would have to be given an initial condition for each state variable in order to obtain a complete solution.