Compute the rank of a matrix up to 5×5 via row echelon form with nullity, rank-nullity theorem verification, pivot visualization, and step-by-step row reduction.
The rank of a matrix is the maximum number of linearly independent rows (or equivalently columns). It tells you the dimension of the image of the linear transformation represented by the matrix and is one of the most fundamental concepts in linear algebra.
Computing rank involves reducing the matrix to row echelon form (REF) or reduced row echelon form (RREF) and counting the number of non-zero rows, which equals the number of pivot positions. This calculator performs complete row reduction with partial pivoting and displays every step of the process.
The rank-nullity theorem states that for an m×n matrix A, rank(A) + nullity(A) = n, where nullity is the dimension of the null space (kernel). This means the number of pivot columns plus the number of free columns always equals the total number of columns. The tool verifies this identity and identifies which columns are pivot columns and which are free.
Rank determines key properties of the matrix: a square matrix is invertible if and only if it has full rank, a system Ax = b is consistent for every b only if A has full row rank, and the solution is unique only if A has full column rank. The calculator shows all these implications with clear visual indicators.
Determining rank requires row-reducing the matrix to echelon form and counting the pivots — a process with many row operations where a single mistake changes the pivot count. This calculator performs full row reduction, identifies pivot and free columns, computes both rank and nullity (via the rank-nullity theorem), and visualizes the pivot structure. It is the fastest way for students to check their row reduction work and for engineers to determine whether a system is solvable or a matrix is full rank.
rank(A) + nullity(A) = n, where n is the number of columns. Rank = number of pivots in RREF.
Result: Rank = 2, Nullity = 1
Row reducing yields 2 pivot rows. The third row becomes all zeros, so rank = 2. By rank-nullity, nullity = 3 − 2 = 1.
The rank of a matrix equals the number of **pivot positions** (leading 1s) in its Row Echelon Form (REF) or Reduced Row Echelon Form (RREF). The algorithm applies elementary row operations — swap, scale, and add multiples — to create zeros below (and optionally above) each pivot. Columns containing pivots are **pivot columns** (linearly independent); columns without pivots are **free columns** (linearly dependent on pivot columns). The rank is also the dimension of the **column space** and the **row space**.
For an m×n matrix A, rank(A) + nullity(A) = n, where nullity is the dimension of the null space (kernel) — the set of solutions to Ax = 0. This theorem partitions the n columns into rank many independent directions and nullity many “lost” directions. A full-rank matrix (rank = min(m,n)) maps no non-zero vector to zero, while a rank-deficient matrix collapses some directions. For square matrices, full rank is equivalent to invertibility.
In **systems of equations**, rank determines solvability: rank(A) = rank([A|b]) is required for consistency, and rank = n (number of unknowns) guarantees a unique solution. In **data science**, the rank of a data matrix indicates how many independent features exist — low-rank approximations (via SVD) are the basis of dimensionality reduction and recommender systems. In **control theory**, the rank of the controllability matrix determines whether a system can be steered to any state. Rank is one of the most fundamental properties of any matrix.
The rank is the maximum number of linearly independent rows or columns. It equals the number of pivot positions in row echelon form.
For an m×n matrix, rank + nullity = n (number of columns). Nullity is the dimension of the null space — the number of free variables in Ax = 0.
A matrix has full rank when rank = min(rows, columns). For square matrices, this means the matrix is invertible.
Pivot columns contain leading 1s in RREF and correspond to determined variables. Free columns have no pivots and correspond to free parameters.
For square matrices, rank = n if and only if det ≠ 0. A rank-deficient square matrix always has determinant zero.
No. Rank ≤ min(m, n) for an m×n matrix. This is because you cannot have more independent rows than total rows, or more independent columns than total columns.