How to use the Matrix Rank Calculator
- 1. Type the matrixOne row per line. Whole numbers, decimals and fractions like 3/4 are fine.
- 2. Press Find rankThe tool reduces the matrix to row echelon form with elementary row operations.
- 3. Count the non-zero rowsThe number of non-zero rows left is the rank. Nullity and full-rank status come with it.
What rank means
The rank ρ(A) of a matrix is the number of linearly independent rows, which equals the number of independent columns. It is found by reducing the matrix to row echelon form with elementary row operations: swapping two rows, multiplying a row by a non-zero number, and adding a multiple of one row to another. These operations never change the rank.
The tool works with exact fractions, so there are no rounding errors in the steps you copy into your notebook.
Worked example
Take A = [1 2 3 0; 2 4 3 2; 3 2 1 3; 6 8 7 5]. In column 1, do R2 → R2 − 2R1, R3 → R3 − 3R1 and R4 → R4 − 6R1. The rows become (0 0 −3 2), (0 −4 −8 3) and (0 −4 −11 5). Column 2 has a zero in row 2, so swap R2 ↔ R3, then R4 → R4 − R2 gives (0 0 −3 2). In column 3, R4 → R4 − R3 makes row 4 all zeros.
Three non-zero rows remain, so ρ(A) = 3. The matrix is 4×4, so it is not full rank, its determinant is 0 and the nullity is 4 − 3 = 1.
Where rank is used
Rank tells you whether a system of linear equations AX = B has solutions. If ρ(A) = ρ([A|B]) = number of unknowns, there is one solution. If both ranks are equal but smaller, there are infinitely many. If they differ, there is none. Enter the augmented matrix to get ρ([A|B]). For eigenvalues of a square matrix, try the eigenvalue calculator; to solve a diagonally dominant system by iteration, use Gauss-Seidel.
Frequently asked questions
Is the rank of a matrix the same for rows and columns?
Yes. Row rank always equals column rank, so you can reduce either way.
Can the rank be more than the number of rows?
No. Rank is at most the smaller of the number of rows and columns.
Does the tool give normal form?
It gives row echelon form, which is enough to read the rank. Normal form needs column operations too and gives the same rank.
Is this calculator free?
Yes. You enter your name and WhatsApp number once when you see your result, and other Uncodemy tools then open without the form.