$(1)$ For matrices $A,B,C$ and $D$ answer the following questions using MATLAB:

$A=\left[\begin{array}{ccccc}3& 1& 4& 1& 5\\ 5& 2& -3& 2& 1\\ -1& 4& 3& -2& 2\\ 3& 0& 5& 5& -1\\ -2& 5& 6& 4& -4\end{array}\right],B=\left[\begin{array}{cc}3& -2\\ 3& 4\\ 6& 4\\ 4& -6\\ -3& 5\end{array}\right],C=\left[\begin{array}{cccc}5& -4& 3& -2\\ 4& -3& -5& 5\\ -4& 2& -2& 1\\ -3& 7& 4& 7\end{array}\right]$

$AB$ if possible

The determinant of matrix $A.$

The inverse of matrix $C.$

Please submit your MATLAB code with code results.

$(2)$ An oscillator is governed by the following equation of motion

$x^{}(t)+x^{}(t)+x^3(t)=$Acos$(t)$

Choose values for $,,$ and $A,$ and answer the following questions:

Write the state space model for this system.

Write an M$-$file using MATLAB to plot the position $x(t)$ and velocity $x^{}(t)$ for this oscillator.