Gaussian Elimination

Enter an augmented matrix, and perform row operations.

Brief Explanation

This page helps you solve systems of linear equations by specifying the coefficients (in the augmented matrix) and performing elementary row operations. This lets you avoid trivial (but frustrating) algebra errors, but still gives you practice at deciding which operations to use.

Of course you can just go to the wolframalpha web site and get the answer, but you do not see the steps involved, and don’t learn anything.

Try It Yourself

  1. Type the augmented matrix in the “Starting Matrix” field. (Click to see the format you should use.)
  2. Click . Your matrix appears, with formatting, in the “Result Log” field. This is where accumulating changes are written.
  3. Enter numbers in one of the lines in “Operations” and click the corresponding button.
  4. If you want to undo a step, click the button.

Your goal, of course, is to get zeroes in the lower-left of the matrix. (“Row-echelon form.”) Then back-solve to leave only 1s in the diagonal. (“Reduced row-echelon form.”)

Starting Matrix

Result Log

• Switch row with row
• Multiply row by
• Divide row by
• Add row to row
• Subtract row from row
• Add × row to row

○ Undo the last step

More Explanation

The operations provided are not the complete set of allowable ones, but they are sufficient to solve any system of equations. Yes, it is redundant to include a “subtract” operation when there is already an “add” operation. But some people just think that way.

It is more justifiable to include separate “multiply” and “divide” operations, since reciprocals may not have terminating decimal representations.

The last operation is most helpful during back-solving, when you already have some variables solved for. You don’t want to re-scale those rows, but use them to cancel out terms in other rows.

To Do

See Also

Some of my other Web pages.