Q&A for How to Reduce a Matrix to Row Echelon Form

At a fundamental level, matrices are objects containing the coefficients of different variables in a set of linear expressions. Each row is for a single expression, and each column is for a single variable. When solving sets of equations, we can combine equations by adding or subtracting the equations, or multiplying them by a factor; it wouldn't make sense to multiply the coefficients of a single variable in all the equations by a number, or subtract the coefficient of one variable from that of another variable in all the equations. Hence, we perform operations on rows (coefficients in expressions), not on columns (coefficients of variables).

The rank of a matrix is the dimension of the vector space spanned by the columns. So the number of pivots equals the rank. The number of non-zero rows also equals the rank.

Yes, but there will always be the same number of pivots in the same columns, no matter how you reduce it, as long as it is in row echelon form. The easiest way to see how the answers may differ is by multiplying one row by a factor. When this is done to a matrix in echelon form, it remains in echelon form.

Just do the row operations until there are 0s below each pivot. It's very simple, just practice it.

One of the applications of reducing to row echelon form is part of the solution of linear equations. The process involves doing the operations described here on the coefficient matrix, while you do the same operations on the vector that corresponds to the right hand side of the equation system. Working with column operations would not really carry over to this application.