The rank is the number of genuinely independent rows in a matrix — equivalently, of independent columns, because the two counts are always the same. Press rank(A) and the answer appears; the step-by-step panel shows the elimination behind it, which is where the number actually becomes meaningful.
Why the number matters
Rank measures how much information a matrix really carries. A 3×3 matrix of rank 3 has three independent directions and is invertible. One of rank 2 has a row that is a combination of the others: it flattens three-dimensional space onto a plane, and no inverse can bring it back.
For a system of equations this is the whole story. Compare the rank of the coefficient matrix with the rank of the augmented matrix: if they differ, the system is inconsistent and has no solution. If they agree and equal the number of unknowns, the solution is unique. If they agree but fall short, the gap is the number of free variables — the dimension of the solution space.
Worked example
Here the third row is exactly the first plus the second, so it contributes nothing new. Elimination drives it to zero and two non-zero rows are left:
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 5 | 7 | 9 |
| 1 | 2 | 3 |
| 0 | −3 | −6 |
| 0 | 0 | 0 |
The calculator reduces the matrix with partial pivoting and counts the rows that survive. Rank is never larger than the smaller of the two dimensions, so a 3×5 matrix can have rank at most 3 — five vectors in three-dimensional space cannot all point in independent directions.
Common mistakes
- Counting non-zero rows before finishing. Rows only reveal themselves as dependent once elimination is complete. Reduce fully, then count.
- Assuming rank equals the number of rows. That holds only when every row is independent — which is exactly the thing being tested.
- Expecting rank(A + B) = rank(A) + rank(B). Rank is subadditive, not additive; adding matrices can destroy independence as easily as create it.
- Rounding decimals by hand. A row that should reduce to zero may leave a residue like 1e−16 and be miscounted. The calculator works in exact fractions to avoid precisely this.