Matrix Rank Calculator

The rank of a matrix is the number of linearly independent rows (or equivalently, columns) it contains. It equals the number of non-zero rows in the matrix after it has been reduced to row echelon form using elementary row operations.

Matrix rank is calculated by applying elementary row operations to transform the matrix into row echelon form. The rank equals the number of non-zero rows (pivot rows) remaining in that reduced form. This process is sometimes called Gaussian elimination.

A matrix is full rank when its rank equals the smaller of its number of rows and columns — that is, rank = min(m, n). For a square matrix, full rank means the matrix is invertible (non-singular) and its determinant is non-zero.

The Rank-Nullity Theorem states that 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 the homogeneous system Ax = 0.

No. The rank of a matrix can never exceed either the number of rows or the number of columns. Formally, rank(A) ≤ min(m, n) for an m×n matrix.

The rank of a zero matrix — a matrix where every entry is 0 — is always 0, because it has no non-zero rows and therefore no linearly independent rows.

For a system of linear equations Ax = b, the rank of the coefficient matrix A determines whether solutions exist and how many there are. If rank(A) equals rank of the augmented matrix [A|b], the system is consistent. If rank(A) equals the number of unknowns, there is exactly one solution.

The three elementary row operations are: (1) swapping two rows, (2) multiplying a row by a non-zero scalar, and (3) adding a scalar multiple of one row to another. These operations preserve the rank of the matrix while reducing it to row echelon form.