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To my understanding, a matrix is just a way of representing a system of equations in an organized format.

So for example, if we have some system of equations, we can get them into standard form, and translate them into what's known as an augmented matrix. This is similar to using synthetic division for dividing polynomials.

Now one rule for elementary matrix operations is that:

Some row, ##j##, can be replace with the sum of itself and a constant multiple of row ##i##- denoted as ##(cRi+Rj)##.

Now my questions is, why doesn't this change the solutions to the system of equations?

Take for example the matrix in my attached picture. We are multiplying ##R_1## by -5, and then adding that result to ##R_2##. Rows 1 and 3 remain the same as when we started, but Row 2 has changed in a way that is

So for example, if we have some system of equations, we can get them into standard form, and translate them into what's known as an augmented matrix. This is similar to using synthetic division for dividing polynomials.

Now one rule for elementary matrix operations is that:

Some row, ##j##, can be replace with the sum of itself and a constant multiple of row ##i##- denoted as ##(cRi+Rj)##.

Now my questions is, why doesn't this change the solutions to the system of equations?

Take for example the matrix in my attached picture. We are multiplying ##R_1## by -5, and then adding that result to ##R_2##. Rows 1 and 3 remain the same as when we started, but Row 2 has changed in a way that is

*not*a multiple of itself in the given matrix. I understand that this process is to get the matrix into row echelon form so that we can perform Gaussian Elimination, but I don't understand why our solutions haven't changed now that we have a different multiple of Row 2.