Calculators
These run on exact fraction arithmetic, so the output looks like hand work rather than floating-point noise. Use them to check your own elimination, not to replace it — every one of them shows its steps.
Row reduction / rank / null space
Exact fraction arithmetic — type entries like 3, -2, or 1/2. One row per line.
| 1 | 2 | -1 | 1 |
| 2 | 4 | -3 | 0 |
| 1 | 2 | 1 | 5 |
| 1 | 2 | 0 | 3 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 |
| -2 |
| 1 |
| 0 |
| 0 |
| -3 |
| 0 |
| -2 |
| 1 |
| 1 |
| 2 |
| 1 |
| -1 |
| -3 |
| 1 |
Determinant & inverse
Reduces to triangular form, tracking every sign flip. Square matrices only.
| 2 | 1 | 3 |
| 4 | 2 | 1 |
| 6 | -3 | 4 |
| -11/60 | 13/60 | 1/12 |
| 1/6 | 1/6 | -1/6 |
| 2/5 | -1/5 | 0 |
Eigenvalues, eigenspaces & diagonalizability
Builds the characteristic polynomial exactly, factors out rational roots, then solves the null space of A − λI.
| 1 |
| 1 |
| 1 |
| 3 |
| 1 |
| 0 |
| -1 |
| 0 |
| 1 |
Gram–Schmidt & QR
Columns of the matrix are the input vectors. The un-normalised vectors stay exact; Q and R are shown numerically because norms bring in square roots.
| 1 |
| 1 |
| 1 |
| 1 |
| -5/2 |
| 5/2 |
| 5/2 |
| -5/2 |
| 2 |
| -2 |
| 2 |
| -2 |
| 0.5 | -0.5 | 0.5 |
| 0.5 | 0.5 | -0.5 |
| 0.5 | 0.5 | 0.5 |
| 0.5 | -0.5 | -0.5 |
| 2 | 3 | 2 |
| 0 | 5 | -2 |
| 0 | 0 | 4 |
Least squares via the normal equations
Enter the augmented matrix (A | b) — one row per equation, last column is b. Everything stays exact because AᵀA and Aᵀb are rational.
| 1 | 1 |
| -2 | 3 |
| 2 | -1 |
| 3 |
| 1 |
| 2 |
| 9 | -7 |
| -7 | 11 |
| 5 |
| 4 |
| 83/50 |
| 71/50 |
| 77/25 |
| 47/50 |
| 19/10 |
If the last column of A is a column of x-powers, these are the coefficients of the best-fit polynomial, in the order the columns appear.
1/2 and decimals like 0.25 both work.