What is the rank of a matrix?

The rank of a matrix is the dimension of its row space or column space. In other words, it is the maximum number of linearly independent rows or columns. For a matrix A, the rank indicates how many independent equations or variables are present in the associated linear system.

How does this calculator compute the matrix rank?

This calculator uses numerical Gaussian elimination to transform the matrix into row echelon form and count the number of non zero rows. It also produces a reduced row echelon form for easier inspection of pivot positions and linear independence.

What is the difference between full row rank and full column rank?

A matrix has full row rank if its rank equals the number of rows, meaning the rows are linearly independent. It has full column rank if its rank equals the number of columns, meaning the columns are linearly independent. A square matrix is full rank when it has both full row rank and full column rank, which implies it is invertible.

Is the computation exact or numerical?

The calculator uses floating point arithmetic with a small tolerance to detect zero pivots. For integer and simple rational matrices this is usually reliable, but for ill conditioned matrices or very large entries numerical effects can appear. For rigorous symbolic rank computations, a computer algebra system is recommended.

What is nullity and how is it related to rank?

The nullity of a matrix is the dimension of its null space, that is the number of independent solutions of Ax = 0. For an m by n matrix, the rank nullity theorem states that rank(A) plus nullity(A) equals n, the number of columns. The calculator reports both rank and nullity.

Matrix Rank Calculator

Gaussian elimination, row echelon form and nullity

Enter any real matrix and let the calculator compute its rank using numerical Gaussian elimination. The tool produces a reduced row echelon form, identifies pivot columns, and classifies the matrix (full rank, singular, full row or column rank).

Designed for linear algebra students, instructors and practitioners who want a transparent and didactic view of matrix rank, not just the final number.

1. Define matrix size and entries

Number of rows (m)

From 1 to 8.

Number of columns (n)

From 1 to 8.

Change m and n, then click to resize the grid. Empty cells are treated as zero.

Matrix A (entries a_ij)

Load example:

Use decimal notation. For integer matrices the algorithm behaves almost exactly like hand Gaussian elimination.

2. Rank, nullity and classification

The algorithm computes the rank of A by counting pivot rows after transforming the matrix into reduced row echelon form. It also reports nullity and whether the matrix has full row or column rank.

Rank(A)

—

dimension of row and column space

Nullity(A)

—

number of free variables (n − rank)

Row / column rank

—

full row rank? full column rank?

Square matrix status

—

invertible or singular (if m = n)

3. Reduced row echelon form (RREF)

The reduced row echelon form makes pivot positions and linear dependencies explicit. Non zero rows correspond to independent rows of A.

RREF(A) will appear here after computation.

Values smaller than a numerical tolerance (currently 1e−10 in absolute value) are displayed as zero.

Rank of a matrix: definition and intuition

In linear algebra, the rank of an \(m \times n\) matrix \(A\) is defined as the dimension of its row space or, equivalently, the dimension of its column space. Intuitively, it tells you how many rows or columns carry independent information.

\[ \operatorname{rank}(A) = \text{dimension of the span of the rows of } A = \text{dimension of the span of the columns of } A. \]

The rank satisfies \(0 \le \operatorname{rank}(A) \le \min(m,n)\). When \(\operatorname{rank}(A) = \min(m,n)\) we say that \(A\) has full rank.

Rank and solutions of linear systems

Given a linear system \(A x = b\), the rank of the coefficient matrix \(A\) and the rank of the augmented matrix \([A | b]\) determine the structure of the solution set:

If \(\operatorname{rank}(A) = \operatorname{rank}([A|b]) = n\), the system has a unique solution.
If \(\operatorname{rank}(A) = \operatorname{rank}([A|b]) < n\), the system has infinitely many solutions.
If \(\operatorname{rank}(A) \ne \operatorname{rank}([A|b])\), the system is inconsistent (no solutions).

The calculator computes only \(\operatorname{rank}(A)\), but you can reason about these cases by extending the matrix with a column \(b\) and recomputing.

How the calculator computes rank (Gaussian elimination)

The rank is computed by transforming \(A\) into a row equivalent matrix in row echelon form and then counting the number of non zero rows. The steps replicated by the algorithm are:

Search for a pivot (a non zero entry) in each column, starting from the top left.
Swap rows if necessary so that the pivot moves to the current row.
Normalize the pivot row so that the pivot becomes 1.
Eliminate the entries below the pivot using row operations.
Repeat for the next column and row until there are no more pivots.

In a second pass, the calculator also eliminates entries above each pivot to produce the reduced row echelon form (RREF).

\[ A \sim R \quad\Rightarrow\quad \operatorname{rank}(A) = \operatorname{rank}(R), \] where \(\sim\) denotes row equivalence and \(R\) is the RREF of \(A\). The number of non zero rows of \(R\) equals \(\operatorname{rank}(A)\).

Rank, nullity and the rank–nullity theorem

For an \(m \times n\) matrix \(A\), the nullity of \(A\) is the dimension of the solution space of the homogeneous system \(A x = 0\). The rank–nullity theorem states that

\[ \operatorname{rank}(A) + \operatorname{nullity}(A) = n. \]

This calculator reports both \(\operatorname{rank}(A)\) and \(\operatorname{nullity}(A) = n - \operatorname{rank}(A)\). A nullity of zero means that the only solution of \(A x = 0\) is the trivial one, so the columns are linearly independent.

FAQ: understanding matrix rank

1. Why can rank be computed from rows or columns?

Row operations do not change the row space dimension, and every row operation can be expressed as multiplication by an invertible matrix on the left. This implies that the dimension of the row space equals the dimension of the column space, so row rank equals column rank.

2. What does it mean for a square matrix to be singular?

A square matrix \(A\) is singular when its rank is less than its size. For an \(n \times n\) matrix, this means \(\operatorname{rank}(A) < n\). In this case \(A\) is not invertible, its determinant is zero, and the linear system \(A x = b\) is either inconsistent or has infinitely many solutions.

3. Can two different matrices have the same rank?

Yes. Rank is only one invariant of a matrix and does not fully describe it. Many matrices can share the same rank but represent different linear transformations. However, matrices with the same rank and size have isomorphic row and column spaces.