Algorithm $3\mathrm{.}1\mathrm{.}6($Gaussian Elimination$)$ Given an $nn$ matrix A and $n-$vector $b,$ the following algorithm reduces the system of linear equations $Ax=b$ to an equivalent upper triangular system. A and $b$ are overwritten with the coefficients and right$-$hand side, respectively, of the reduced system.

for $j=1,2,$dots,$n-1$ do

for $i=j+1,j+2,$dots,$n$ do

$m_{ij}=\frac{a_{ij}}{a_{jj}}$

for $k=j+1,j+2,$dots,$n$ do

$a_{ik}=a_{ik}-m_{ij}{a}_{jk}$

end for

$b_i=b_i-m_{ij}{b}_j$

end for

end for

Exploration $3\mathrm{.}1\mathrm{.}7$ Write a MATLAB function

$[A,b]=$gausse$\underset{?}{lim}(A,b)$

that implements Algorithm $3\mathrm{.}1\mathrm{.}6.$