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A matrix's rank tells you how many of its rows (or columns) are truly independent — a key concept for understanding whether a system of equations has a unique solution.
This calculator finds the rank of any matrix — square or rectangular, with rows and columns each from 2 up to 6 — by reducing it to row echelon form.
How it works
Enter the number of rows and columns and the matrix elements — rows and columns don't need to match. The calculator uses Gauss-Jordan elimination (row swaps, scaling, and row combinations) to reduce the matrix to row echelon form, then counts the number of non-zero rows remaining — that count is the rank.
- Click Calculate to see your results.
Examples
A dependent 2×2 example
The matrix [[1,2],[2,4]] has rank 1, since its second row is just twice the first — the rows aren't independent.
Who should use it
- Checking whether a system of linear equations has a unique solution.
- Coursework involving linear independence and vector spaces.
Industry applications
- Linear algebra and mathematics education
- Engineering and data analysis
Advantages
- Shows the full row echelon form used to determine the rank.
- Works for both square and rectangular matrices, up to 6 rows and 6 columns.
Limitations
- Capped at 6 rows and 6 columns to keep computation instant on a public, synchronous endpoint.
Common mistakes to avoid
- Assuming every square matrix automatically has full rank — many don't, particularly if one row is a multiple of (or combination of) others.
- Confusing rank with the number of rows or columns — rank measures independence, not size.
Best practices
- If you're checking whether a system of equations has a unique solution, compare the coefficient matrix's rank to its size — they must match for a unique solution to exist.
Tips
- A matrix has full rank if and only if its determinant is nonzero (for square matrices) — if you already know the determinant is zero, the rank must be less than the matrix size.