Everyday math
Matrix Calculator
Add, multiply, invert, and row reduce matrices with exact fractions. See every row operation that gets you to RREF.
Matrices
Type whole numbers, decimals, or fractions like 2/3. Decimals become exact fractions. Enter moves to the next cell.
Calculate
Try 2A−B, A*B, A^-1, A^T, det(A), rref(A), trace(A), or rank(A).
3 × 3
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
Row Steps to RREF9 steps
Each step uses one row operation. The highlighted entry is the pivot being worked on.
Start
2 1 −1 −3 −1 2 −2 1 2 1Multiply row 1 by 1/2.Scale the row so its pivot is 1.
1 12 −12 −3 −1 2 −2 1 2 2Add 3 times row 1 to row 2.Clear the entry below the pivot.
1 12 −12 0 12 12 −2 1 2 3Add 2 times row 1 to row 3.Clear the entry below the pivot.
1 12 −12 0 12 12 0 2 1 4Multiply row 2 by 2.Scale the row so its pivot is 1.
1 12 −12 0 1 1 0 2 1 5Subtract 1/2 times row 2 from row 1.Clear the entry above the pivot.
1 0 −1 0 1 1 0 2 1 6Subtract 2 times row 2 from row 3.Clear the entry below the pivot.
1 0 −1 0 1 1 0 0 −1 7Multiply row 3 by -1.Scale the row so its pivot is 1.
1 0 −1 0 1 1 0 0 1 8Add row 3 to row 1.Clear the entry above the pivot.
1 0 0 0 1 1 0 0 1 9Subtract row 3 from row 2.Clear the entry above the pivot.
1 0 0 0 1 0 0 0 1
How to Use
Build Your Matrices, Then Pick an Operation
Each matrix has a letter. Use the plus and minus buttons to set its rows and columns, then type the entries. Tap a matrix letter to make it the one the operation buttons use. The result updates as you type, and Save as New Matrix keeps a result so you can use it in the next step.
Type Your Own Expression
- Add and subtract: A+B, A−B
- Multiply: A*B or AB, and a number times a matrix: 3A
- Transpose and inverse: A^T and A^-1, or inv(A)
- Powers: A^2, A^3
- Numbers from a matrix: det(A), trace(A), rank(A)
- Row reduce: rref(A), with every row operation shown
For a single number, the Scientific Calculator handles trig, logs, and roots.
Questions
Why do my answers show fractions?
The calculator keeps every number exact, so 0.1 is stored as 1/10 and an inverse shows 3/5 instead of 0.6. Tap Decimals to see the same result as decimals.
What does it mean when a matrix has no inverse?
A square matrix has an inverse only when its determinant is not 0. When it is 0, at least one row can be built from the other rows, so row reducing it leaves a row of zeros. That is called a singular matrix.
When can I multiply two matrices?
A×B works when A has as many columns as B has rows. A 2×3 matrix times a 3×4 matrix gives a 2×4 result. Order matters, so A×B and B×A are usually different.
What is RREF?
Reduced row echelon form is what you get after row reducing as far as possible: each pivot is 1, and every other entry in a pivot column is 0. Open the row steps to see each operation, from swapping rows to clearing entries.
Can I type square roots or pi into a cell?
Yes. Cells accept expressions like sqrt(2) or pi/4. Those are not exact fractions, so results that use them are shown as decimals.