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I for the most part understand the idea of the pivot column, but I'm slightly confused. I know a pivot is the leading coefficient in a column with all the rows below containing 0's. However do the pivots have to be along the diagonal? For example here the only pivot column would be the one containing the two, correct?

\begin{matrix} 2 & 3 & \\ 0 & 0 & \\ 0 & 0 & \\ \end{matrix}

Also for here only the column with the two would be a pivot column? \begin{matrix} 2 & 3 & 6 \\ 0 & 0 & 0 \\ \end{matrix}

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In your examples, only the first column is a pivot column, just as you say. Your definition doesn’t feel quite right to me, though. A pivot is the first non-zero element in a row. A pivot column is then a column that has a pivot in it. They don’t have to be along the diagonal.

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Do the pivots have to be along the diagonal?

No. Here's a matrix with a pivot off the diagonal.

\begin{matrix} 2 & 3 & 6 \\ 0 & 0 & 1 \\ \end{matrix}

Here both the 2 and the 1 are in pivot positions because they correspond to a leading 1 in the reduced echelon form of the matrix.

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