$x_1-5x_2=5$

$3x_1-x_2+4x_3=-11$

$x_1+10x_2+x_3=10$

$[$Solve in computer$]$ Use a programming language $($python or R$)$ to solve for question $(1)$ in computer as follows. Make sure you have pracma library in R$,$ and numpy, scipy and sympy libraries in Python.

a$)$ Define the coefficient matrix and right$-$hand$-$side vector in your code for the problem

b$)$ Form the augmented matrix $($Use $np.$append in python, cbind in R$)$

c$)$ Apply rref $($reduced row echelon form$)$ function to the augmented matrix to apply Gauss$-$Jordan elimination. Now you have the solution as the last column of the processed augmented matrix.

d$)$ Use $lu$ function to identify $L$ and $U$ matrices necessary for $LU$ decomposition based solution. Solve $Ly=b$ and $Ux=y$ equations by applying rref function twice $($Gauss Jordan elimination$).$

e$)$ Use matrix$-$inverse based solution $)=(A^{-1}b$ to solve for the unknowns. But, do not use inv function, instead, use rref to get the inverse of A matrix.

f$)$ Calculate determinant and rank of the coefficient matrices. What do they tell you? $[$Hint: you can use rank and det functions in R$,$ det and matrix$\_$rank functions in numpy$]$