Can all equation systems be reduced to the identity matrix? I'm trying to learn about solving equation systems using the Gauss-Jordan method. So, you have to convert the equation system to a matrix, and then reduce it to the identity. When you transform it to the augmented matrix, it is pretty easy to find the solutions by going backwards...


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*Can all equation systems be reduced to the identity matrix? If not, how do you determine whether you can or not?


And, as a side note:


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*From my course, it would seem like this is pretty much a "lightbulb" technique. You literally stare at the matrix and suddenly it occurs to you how to proceed. Is that correct? Is there really no standard?

 A: Not all systems of linear equations can be reduced to the identity matrix. A (square) matrix can only be reduced to the identity if the matrix is invertible (i.e., has an inverse). There are many ways to check if a matrix is invertible; some ways include checking if the determinant is non-zero, or if for an $n\times n$ matrix, the rank is $n$.
I wouldn't really say the Gauss-Jordan algorithm is a "lightbulb" technique. It's an algorithm, so you just keep repeating it until the matrix is in (Reduced) Row Echelon Form. 
Given for example, \begin{bmatrix}
2&4&2\\
0&2&3\\
3&2&1
\end{bmatrix}
You start with the $a_{11}=2$ and make it $1$ by multiplying the first row by $\frac{1}{2}$.
$\to \begin{bmatrix}
1&2&1\\
0&2&3\\
3&2&1
\end{bmatrix}$
Then you want to make sure $a_{21}=0$. It is already $0$, so we are in luck. So let's move on. We want $a_{31}=1$ next; it's a $3$, so let's multiply row $1$ by $3$ and subtract row $3$.
$\to \begin{bmatrix}
1&2&1\\0&2&3\\0&4&2
\end{bmatrix}$.
Then you want to make $a_{22}=1$ by multiplying row $2$ by $\frac{1}{2}$. And you continue this procedure.
Essentially, you multiply the row by some constant in order to make the term you want into a $1$, and then multiply that $1$ by some constant and add or subtract some row until the numbers below and above it are $0$, just like how we did it for the first column.
