Enter an augmented matrix, and perform row operations.

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.

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

Operations

• 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

• Multiply row by

• Divide row by

• Add row to row

• Subtract row from row

• Add × row to row

○ Undo the last step

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.

- More error checking. You can definitely mess things up, by e.g. specifying a row number that doesn’t exist.
- Add matrix inversion.
- Much much more...

Some of my other Web pages.

- The Golden Ratio Pages — A series of pages exploring the famous ratio.
- Gear Ratios — These may not be golden, but they are important for people who ride and work with bicycles. Available as a mobile app and a Web page.
- Anagrams — A mobile app and a Web tool for finding anagrams.
- The Chaos Pages — A similar series of pages exploring iterated systems.
- Combinatorial Music Theory — A lecture connecting graph theory with musical scales and chords.
- The 3D Pages — My JavaScript implementation of interactive 3D graphics .
- The DSP Pages — Explaining the Fourier transform in the discrete domains.
- Graph Clock — A good example of using JavaScript to make a self-modifying Web page, and a little puzzle about elementary connected graphs.
- Regular Expressions — Sometimes a non-match can hang the system.
- The Z-Board — A new kind of MIDI controller.