Download Article

Download Article

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.

Steps

  1. 1

    Understand what row-echelon form is. The row-echelon form is where the leading (first non-zero) entry of each row has only zeroes below it. These leading entries are called pivots, and an analysis of the relation between the pivots and their locations in a matrix can tell much about the matrix itself. An example of a matrix in row-echelon form is below.[1]

    • ( 1 1 2 0 1 3 0 0 5 ) {\displaystyle {\begin{pmatrix}1&1&2\\0&1&3\\0&0&5\end{pmatrix}}}

  2. 2

    Understand how to perform elementary row operations. There are three row operations that one can do to a matrix.[2]

    • 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.

    Advertisement

  3. 3

    Begin by writing out the matrix to be reduced to row-echelon form. [3]

    • ( 1 1 2 1 2 3 3 4 5 ) {\displaystyle {\begin{pmatrix}1&1&2\\1&2&3\\3&4&5\end{pmatrix}}}

  4. 4

    Identify the first pivot of the matrix. The pivots are essential to understanding the row reduction process. When reducing a matrix to row-echelon form, the entries below the pivots of the matrix are all 0.[4]

    • 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.

  5. 5

    Perform row operations on the matrix to obtain 0's below the first pivot. [5]

  6. 6

    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.

  7. 7

    Perform row operations on the matrix to obtain 0's below the second pivot.

  8. 8

    In general, keep identifying your pivots. Row-reduce so that the entries below the pivots are 0.

    9. Advertisement

Add New Question

  • Question

    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).

  • Question

    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.

  • Question

    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.

See more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Advertisement

Video

  • Reducing a matrix to row-echelon form works with any size matrix, both square and rectangular.

Thanks for submitting a tip for review!

Advertisement

References

About This Article

Thanks to all authors for creating a page that has been read 218,094 times.

Did this article help you?