How to Row‐Reduce Matrices

Reducing Matrices to Row Echelon Form

Row-reducing matrices to a row echelon form is an important calculation in linear algebra that can be useful for solving systems of equations. This procedure allows users to express a linear system of equations as an upper-triangular matrix. Follow these steps to row reduce a matrix to its row echelon form.

1. Start by writing down the matrix to be row-reduced. Matrix A is the example for this example.
   
   A =
   $\begin{bmatrix} 2 & 3 & 6 \\ 4 & 9 & 18 \\ 6 & 12 & 24 \\ \end{bmatrix};$
2. Choose an elementary row operation to perform on the matrix. In this case, we will use row substitution.
   
   To perform a row substitution, multiply row 1 by a factor f and then add it to row 2.
   
   f = (Row 2 Escape Column 1)/(Row 1 Escape Column 1)
   
   f = (4)/(2) = 2
   
   2 * Row 1 + Row 2 ---> Row 2
3. Compute the new values of row 2.
   
   Row 2 = 8 3 18
4. Choose the next elementary row operation to perform on the matrix. In this case, we will use row substitution again.
   
   To perform a row substitution, multiply row 1 by a factor f and then add it to row 3.
   
   f = (Row 3 Escape Column 1)/(Row 1 Escape Column 1)
   
   f = (6)/(2) = 3
   
   3 * Row 1 + Row 3 ---> Row 3
5. Compute the new values of row 3.
   
   Row 3 = 18 9 24
6. Finally, the matrix has been row reduced to its row echelon form.
   
   A =
   $\begin{bmatrix} 2 & 3 & 6 \\ 8 & 3 & 18 \\ 18 & 9 & 24 \\ \end{bmatrix}$

Step-by-Step Guide to Row Reducing Matrices

A matrix can be row-reduced to put it into its simplest form. This is important for many linear algebra operations, including solving a system of linear equations. Follow the below instructions to learn how to row-reduce a matrix.

Step 1: Identify the Leading Coefficient

The leading coefficient is the left-most, highest-up number in the matrix. If the leading coefficient is a 1, you can move to the next step. If the leading coefficient is anything other than a 1, divide all of the numbers in that row by the leading coefficient.

Step 2: Make All the Numbers below the Leading Coefficient Equal to Zero

Once you identify and divide the leading coefficient's row, use the same number to subtract from each of the rows below it in order to make the leading coefficients of each row underneath the original zero.

Step 3: Check for a Row of Zeroes at the Bottom

If a Row of Zeroes appears at the Bottom, your matrix has been successfully row-reduced! If a row of zeroes does not appear, you will need to repeat the process with the row below the first one.

Step 4: Repeat the Above Steps

Repeat Steps 1–3 with each sub-matrix that is created after Steps 1 and 2, until you have a row of zeroes at the bottom of your matrix. This is the simplest form of the matrix, and is its row reduced form.