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I have a matrix representing a linear combination of 12 frequencies, augmented with a frequency that is always a simple addition of two of the before. enter image description here

For example, 754 from row one is 1*493 + 1*261, so a linear combination of those two frequencies, and a very simple one for now. Every row is the same as that.

That said, I must be misremembering my linear algebra class, as when I RREF it, it blows up:

enter image description here

Am I misremembering the method to use here, or am I rref-ing incorrectly here?

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    $\begingroup$ What is it you believe row reduction does? You are solving a system of simultaneous linear equations, so you are looking for values that make all equations simultaneously true. You say the first row is such that the last column equals second plus third; but as this is not true of the second row, that (second and third variables equal to $1$, all the rest equal to $0$) is not a solution. Your Row reduction shows that there is no solution that solves all rows simultaneously: your system is inconsistent. Again: what were you trying to obtain through row reduction? $\endgroup$ Dec 9, 2021 at 6:36
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    $\begingroup$ It is plain the system corresponding to this matrix is inconsistent, since you have the same equation (indicated by all but the last column) equal to 7 different things. It's as inconsistent as a system that says $x+y=1$ and $x+y=2$. $\endgroup$ Dec 9, 2021 at 6:37

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