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Reducing a Matrix to Row Echelon Form
Using row operations to reduce a matrix to row echelon form is a useful way to solve linear systems of equations. This form simplifies solving because equations become much easier to read and manipulate. With the following steps you can reduce a matrix to row echelon form.
Step 1:
Swap rows to move any leading entries (the first non-zero entries in each row) to the left edge, further to the left than all previous rows. Continue until all rows have leading entries.
Step 2:
Divide the leading entries (the entries on the left edge of the matrix) by themselves. This will make the leading entries of each row equal to 1.
Step 3:
Subtract from each other row, the multiple of the row with the leading one. The multiple is equal to the leading entry of the other row, divided by the leading entry of the original row. This creates a 0 in the leading entry of each of the other rows below the original.
Step 4:
Repeat the process of Step 2 and Step 3 for the columns below the leading entry of the first column. This creates a 0 in all entries below the leading entries.
Step 5:
Repeat the process described in Step 1 to Step 4 for all columns, until all rows are in row echelon form.
Tips:
- Leading entries should appear further to the left than previous rows.
- Leading ones should appear in all columns, if not, you have not completed the row echelon form.
- Row operations should help the last step.