Why is this method for solving linear equations systems using determinants works

linear algebrasystems of equations

I'm new here and english is not my native language but I'll try to explain my question the best I can.
While studying methods for solving systems of linear equations I came across a method so called "Gauss algorithm" for which I doubt if it is it's real name or not, but I can't understand WHY it works.
I would really appreciate a proper explanation or maybe a hint for what is happening inside the method process that accounts for the logic behind it.

The process in question is the following and involves determinants:

Suppose we have the following system of linear equations, for instance
\begin{cases}
\ x+2y-z=-5 \\2x-1y+2z=8 \\3x+3y+4z=5 \
\end{cases}

Then the result of appliying the method continues as follow

\begin{array}{ccc|c}
x & y & z & i.t. \\
\hline
1 & 2 & -1 & -5 \\
2 & -1 & 2 & 8 \\
3 & 3 & 4 & 5 \\
\hline
& -5 & 4 & 18 \\
& -3 & 7 & 20 \\
\hline
& & -23 & -46 \\
\end{array}

Where for example the coefficients $-5$ and $4$, and the independent term $18$, of the first row of the reduced system has been computed with the follow determinants:

\begin{equation}
\begin{vmatrix}
1 & 2 \\
2 & -1 \\
\end{vmatrix}=-5
\hspace{1cm}
\begin{vmatrix}
1 & -1 \\
2 & 2 \\
\end{vmatrix}=4
\hspace{1cm}
\begin{vmatrix}
1 & -5 \\
2 & 8 \\
\end{vmatrix}=18
\end{equation}

Where those determinants were computed using the first and the second row of the system of equations matrix. Then the other coefficients -3 and 7 and the independant term 20 were found by computing the respective determinants as above but using the first and third row of the system of equations matrix.

Appliying this method and if the system is compatible, then as above you can see that \begin{equation} -23z = -46\\
z=2\end{equation}
and then by substitution in the subsequent former equations the other variables can be found.

Well that's the method I was studying and will appreciate to know more about it, it's a pity I don't know the name of it but maybe you can help.

Thanks in advance!

Best Answer

The computation of the determinants is equivalent to solving step by step the system of equations in the following way:

  1. Multiply the first equation by $2$, which is the coefficient of $x$ in the second equation;
  2. Multiply the second one by $1$, which is the coefficient of $x$ in the first equation; and then
  3. Subtract the first from the second equation.

This will result in the fourth equation.

Afterwards we can do the same for the first and third equations to get the fifth equation:

  1. Multiply the first equation by $3$, which is the coefficient of $x$ in the third equation;
  2. Multiply the third one by $1$, which is the coefficient of $x$ in the first equation; and then
  3. Subtract the first from the third equation.

Now we can do a similar approach to the fourth and fifth equation to get the sixth equation:

  1. Multiply the fourth equation by $-3$, which is the coefficient of $y$ in the fifth equation;
  2. Multiply the fifth one by $-5$, which is the coefficient of $y$ in the fourth equation; and then
  3. Subtract the fourth from the fifth equation.

And voilà, we have the sixth equation.

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