# Laplace transform method (using matrices)

#### u-will-neva-no

Joined Mar 22, 2011
230
Hi This is the question I am facing. I will then show my solution up to the point that I am stuck on:

Question: Consider the matrix

$$A=\left{ \begin{array}{lml} 0 & \, & 1\\ 1 & \, & 2\\ \end{array} \right\}$$
Determine exp(At), by using the Laplace transform method.

Current solution:
I used the formula
$$SI - A$$ so:
$$s\left{\begin{array}{lml} 1 &\, & 0\\ 0 &\, & 1\\ \end{array}\right\} - \left{\begin{array}{lml} 0 &\, & 1\\ -1 &\, & -2\\ \end{array}\right\}$$

This comes to:
$$SI -A = \left{\begin{array}{lml} s &\, & -1\\ 1 &\, & s+2\\ \end{array}\right\}$$

Finding the inverse gives:
$$(SI -A)^-^1 =\frac{1}{s(s+2)-(-1*1)} \left{\begin{array}{lml} s+2 &\, & 1\\ -1 &\, & s\\ \end{array}\right\}$$

This simplifies to:
$$(SI -A)^-^1 =\frac{1}{(s+1)^2} \left{\begin{array}{lml} s+2 &\, & 1\\ -1 &\, & s\\ \end{array}\right\}$$

or:

$$(SI -A)^-^1 =\left{\begin{array}{lml} \frac{s+2}{(s+1)^2} &\, & \frac{1}{(s+1)^2}\\ \frac{-1}{(s+1)^2} &\, & \frac{s}{(s+1)^2}\\ \end{array}\right\}$$

I am not sure what to do next. I know that I need to take the Laplace of the final expression but I don't know how to do that... Any help, other examples would help alot!!

#### Peytonator

Joined Jun 30, 2008
105
You're wanting the transition matrix e^At, right? You need to first split those factors inside your final matrix into partial fractions, then take the inverse laplace transform of the whole matrix (i.e. use the laplace transform tables to find the inverse transform of each term)...

• u-will-neva-no
Thanks Peytonator, that helped me out! 