How to Reduce a Matrix to Row Echelon Form

This article was co-authored by Mario Banuelos, PhD. Mario Banuelos is an Associate Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels.

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The row-echelon form of a matrix is highly useful for many applications. For example, it can be used to geometrically interpret different vectors, solve systems of linear equations, and find out properties such as the determinant of the matrix.

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Row swapping.
Scalar multiplication. Any row can be replaced by a non-zero scalar multiple of that row.
Row addition. A row can be replaced by itself plus a multiple of another row.

( 1 1 2 1 2 3 3 4 5 ) 1&1&2\\1&2&3\\3&4&5\end>>

For our matrix, the first pivot is simply the top left entry. In general, this will be the case, unless the top left entry is 0. If this is the case, swap rows until the top left entry is non-zero.
By their nature, there can only be one pivot per column and per row. When we selected the top left entry as our first pivot, none of the other entries in the pivot's column or row can become pivots.

For our matrix, we want to obtain 0's for the entries below the first pivot. Replace the second row with itself minus the first row. Replace the third row with itself minus three times the first row. These row reductions can be succinctly written as R 2 → R 2 − R 1 \to R_-R_> and R 3 → R 3 − 3 R 1 . \to R_-3R_.>
( 1 1 2 0 1 1 0 1 − 1 ) 1&1&2\\0&1&1\\0&1&-1\end>>

Identify the second pivot of the matrix. The second pivot can either be the middle or the middle bottom entry, but it cannot be the middle top entry, because that row already contains a pivot. We will choose the middle entry as the second pivot, although the middle bottom works just as well. [6] X Research source

R 3 → R 3 − R 2 \to R_-R_>
( 1 1 2 0 1 1 0 0 − 2 ) 1&1&2\\0&1&1\\0&0&-2\end>>
This matrix is in row-echelon form now.

In general, keep identifying your pivots. Row-reduce so that the entries below the pivots are 0. [7] X Research source

Community Q&A

Why aren't column operations used to reduce a matrix in its echelon form? Community Answer

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).

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How do you find the rank of a matrix by row echelon form? Community Answer

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.

Can my answer in row echelon form differ? Community Answer

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.