The rank tells you how much independent information a matrix contains: whether a system of equations has a unique solution, whether vectors span a space and how many dimensions a transformation preserves.
How to use the Matrix Rank Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The matrix is reduced to reduced row echelon form; the rank is the number of non-zero rows (pivots).
Formula
rank(A) = number of pivots in rref(A) = dim Col(A) = dim Row(A)
Worked example
Inputs — Matrix size (rows × columns): 3 × 4; a₁: 1; b₁: 2; c₁: 3; d₁: 4; a₂: 2; b₂: 4; c₂: 6; d₂: 8; a₃: 0; b₃: 1; c₃: 1; d₃: 2.
Results — Rank = 2; Full rank? = Rank deficient; Reduced row echelon form = ⎡ 1 0 1 0 ⎤
⎢ 0 1 1 2 ⎥
⎣ 0 0 0 0 ⎦.
Key terms
- Rank
- The dimension of the row or column space.
Frequently asked questions
What is full rank?
Rank equal to the smaller of the number of rows and columns.
Can the rank exceed the number of rows?
No – it is at most min(rows, columns).
Does row reduction change the rank?
No – elementary row operations preserve it.
What is the rank of the zero matrix?
0.