Solve the system of equations $$x_1+10x_2-x_3=3,$$$$2x_1+3x_2+20x_3=7,$$$$10x_1-x_2+2x_3=4$$ using the Gauss-Elimination with partial pivoting.
I tried solving the problem as follows:
We have the augmented matrix as $(A,b)=\begin{pmatrix} 1 && 10 && -1 && 3\\ 2 && 3 && 20 && 7\\ 10 && -1 && 2 && 4\end{pmatrix}.$
We now proceed to do some elementary row operations as follows:
$\begin{pmatrix} 1 && 10 && -1 && 3\\ 2 && 3 && 20 && 7\\ 10 && -1 && 2 && 4\end{pmatrix}$ is the original matrix. We let $R_1,R_2,R_3$ as the three rows of the matrix. The transformations $R_2-2R_1,R_3 -10R_1$ gives us the matrix $\begin{pmatrix} 1 && 10 && -1 && 3\\ 0 && -17 && 22 && 1\\ 0 && -101 && 12 && -26\end{pmatrix}.$ Now, we perform the transformations $R_3-\frac{101}{17}R_2$ to get the matrix $\begin{pmatrix} 1 && 10 && -1 && 3\\ 0 && -17 && 22 && 1\\ 0 && 0 && -118.7 && -144.7\end{pmatrix}.$ This means that the matrices : $$\begin{pmatrix} 1 && 10 && -1 && 3\\ 0 && -17 && 22 && 1\\ 0 && 0 && -118.7 && -144.7\end{pmatrix}$$ and $$\begin{pmatrix} 1 && 10 && -1 && 3\\ 2 && 3 && 20 && 7\\ 10 && -1 && 2 && 4\end{pmatrix}$$ are row equivalent and has the same set of solutions. The matrix $$\begin{pmatrix} 1 && 10 && -1 && 3\\ 0 && -17 && 22 && 1\\ 0 && 0 && -118.7 && -144.7\end{pmatrix}$$ provides us with the following linear system:$$x_1+10x_2-x_3=3,$$$$-17x_2+22x_3=1,$$$$-118.7x_3=-144.7$$ which is indeed equivalent to the given system of equations in the problem. The solutions of the system are as follows: $x_3=\frac{-144.7}{-118.7} =1.219,x_2=\frac{1-22x_3}{-17}=1.518,x_1=3-10x_2+x_3=-10.961.$
So, the solutions are $x_1=-10.961,x_2=1.518,x_3=1.219.$
However the set of values $x_1=-10.961,x_2=1.518,x_3=1.219,$ satisfies the equations $x_1+10x_2-x_3=3$ and $2x_1+3x_2+20x_3=7.$ But it does not satisfy the equation $10x_1-x_2+2x_3=4.$ This means in some part of the solution there might be an error. The answer given is, $x_1=0.37512,x_2=0.28940,x_3= 0.26908.$ Unfortunately, I am not being able to find any mistake in my calculations. Any help regarding this issue will be greatly appreciated.