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When proving the consistency or inconsistency of a system of equation using matrices:

I'm confused about when to operate my matrix rows into Row Echelon Form or Reduced Row Echelon Form.

I'm asking because I've seen different YouTube videos where they prove the consistency or inconsistency of equations by row operating to row echelon form and in some, by row operating to reduced row echelon form.

Please can anyone clarify when to use either of them?

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I think either form will suffice.

The consistency of a system is determined by whether or not your system has a solution. If it doesn't, the system is said to be inconsistent.

For example, consider the matrix $\\A = $\begin{bmatrix}1&1&1\\0&1&1\\0&0&0\end{bmatrix}.

This matrix is in row echelon form, but not reduced but just like its reduced form $B=$

\begin{bmatrix}1&0&0\\0&1&1\\0&0&0\end{bmatrix}

for the vector $b=$ \begin{bmatrix}1\\1\\1\end{bmatrix}

Neither $Ax=b$ or $Bx=b$ has solution, because the last row of the two matrices is zero. (so the system $Ax=b$ or $Bx=b$ are both inconsistent).

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