$(30$p$)$ Let $A$ be an $nn$ matrix with non$-$zero determinant and $b$ be a vector of size $n.$ Then, there is only a unique solution vector $x^{**}$ to linear equation set $Ax=b.$ Assume that all elements of matrix A and vector $b$ are independent continuous uniform random variables between $0$ and $1.$ Under this assumption, on the average, what fraction of the elements of $x^{**}$ would be negative if:

$n=3?$

$n=5?$

$n=10?$

Develop a program where you can use classes MCModel$()$ and MCSim$()$ as they are developed in IE $246$ class. Develop a child class Axb$()$ to MCModel $()$ that overrides the inherited $.$ run $()$ and $.$ reset $()$ methods and simulates the above experiment. Use Monte Carlo simulation to estimate each average fraction and print your results to the console.