Write a function that implements Gaussian elimination to solve a system of equations $(Ax=b)$ by

performing a forward sweep and a back substitution. Input to the function should be A and $b,$ where $A$ is an

$n$ by $n$ matrix and $b$ is the vector of constants $b.$ The output will be the solution vector $x.$ Make sure you

check for the possibility of a zero in the pivot element and subsequently make the appropriate change to

your system so that the problem can still be solved. Assume that A is full$-$rank and you only have one

solution to $Ax=b.$ Be sure to demonstrate that your function works for an example $($theoretical$)$ problem.

$[$Note: A matrix is full$-$rank if its rank is the highest possible for a matrix of the same size.$]$