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!!