Matrix Rank Calculator
Find the rank of any 2x2, 3x3, or 4x4 matrix using Gaussian elimination. This calculator performs row reduction to echelon form and counts the non-zero rows to determine the rank.
The rank of a matrix is the number of linearly independent rows (or equivalently, columns). It tells you the dimension of the vector space spanned by the matrix's rows.
How to Find the Rank
- Write the matrix.
- Use row operations (swap rows, multiply a row by a scalar, add a multiple of one row to another) to reduce it to row echelon form (REF).
- Count the non-zero rows. That count is the rank.
Worked Example
Matrix: [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
Row reduce:
- R2 = R2 - 4R1: [0, -3, -6]
- R3 = R3 - 7R1: [0, -6, -12]
- R3 = R3 - 2R2: [0, 0, 0]
REF: [[1, 2, 3], [0, -3, -6], [0, 0, 0]]
Two non-zero rows, so rank = 2. This means the three rows are not all independent; the third row is a linear combination of the first two.
Key Concepts
- Full rank: A square n x n matrix has full rank if rank = n. This means it is invertible and its determinant is non-zero.
- Nullity: The dimension of the null space = number of columns minus rank. Rank + nullity = number of columns (rank-nullity theorem).
- Rank and systems of equations: A system Ax = b has a solution only if rank(A) = rank([A|b]).