I got
s = -1
1 = -A + B -5C + 5/2
s = 2
1 = 8A + 4B + 4C + 1
then not sure what to do from here.
I got det A = $\displaystyle \begin{bmatrix}1/2&1&1\\-3/2&-1&1\\0&8&4\end{bmatrix}.$= (1/2)(-1)(4)+(1)(1)(0)+(1)(-3/2)(8)-(1)(-1)(0)-(1)(-3/2)(4)-(1/2)(1)(8)
divide by $\displaystyle \begin{bmatrix}1&1&1\\-5&-1&1\\4&8&4\end{bmatrix}.$ = (1)(-1)(4)+(1)(1)(4)+(1)(-5)(8)-(1)(-1)(4)-(1)(-5)(4)-(1)(1)(8) = -12/-24
det C = $\displaystyle \begin{bmatrix}1/2&1&1\\-3/2&-5&1\\0&4&4\end{bmatrix}.$= (1/2)(-5)(4)+(1)(1)(0)+(1)(-3/2)(4)-(1)(-5)(0)-(1)(-3/2)(4)-(1/2)(1)(4)
divide by $\displaystyle \begin{bmatrix}1&1&1\\-5&-1&1\\4&8&4\end{bmatrix}.$= (1)(-1)(4)+(1)(1)(4)+(1)(-5)(8)-(1)(-1)(4)-(1)(-5)(4)-(1)(1)(8) = -12/-24
det D = $\displaystyle \begin{bmatrix}1/2&1&1\\-3/2&-5&-1\\0&4&8\end{bmatrix}.$= (1/2)(-5)(8)+(1)(-1)(0)+(1)(-3/2)(4)-(1)(-5)(0)-(1)(-3/2)(8)-(1/2)(-1)(4)
divide by $\displaystyle \begin{bmatrix}1&1&1\\-5&-1&1\\4&8&4\end{bmatrix}.$= (1)(-1)(4)+(1)(1)(4)+(1)(-5)(8)-(1)(-1)(4)-(1)(-5)(4)-(1)(1)(8) = -12/-24
also the answers say the laplace transforms are -1/2e^tcost + 1/2 + 1/2t
You're not computing the correct determinants. You should have
$\displaystyle \displaystyle A=\frac{\left|\begin{matrix}
1/2&1&1\\
-3/2&-1&1\\
0&8&4
\end{matrix}\right|}{\left|\begin{matrix}
1&1&1\\
-5&-1&1\\
4&8&4
\end{matrix}\right|},
C=\frac{\left|\begin{matrix}
1&1/2&1\\
-5&-3/2&1\\
4&0&4
\end{matrix}\right|}{\left|\begin{matrix}
1&1&1\\
-5&-1&1\\
4&8&4
\end{matrix}\right|},
D=\frac{\left|\begin{matrix}
1&1&1/2\\
-5&-1&-3/2\\
4&8&0
\end{matrix}\right|}{\left|\begin{matrix}
1&1&1\\
-5&-1&1\\
4&8&4
\end{matrix}\right|}.
$
What does that give you?