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!!
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!!