Advanced Rank Theorem Calculator

Enter matrix values

Matrix A

Use spaces, commas, or semicolons between entries.

Optional vector b

Provide one value for each matrix row.

Pivot tolerance

Show row reduction steps

The calculator treats the matrix as a linear map from Rⁿ to R^m.

Example data table

This worked example uses the default matrix shown in the form.

Example matrix A	Rows	Columns	Rank	Nullity	Interpretation
[1 2 3; 2 4 6; 1 1 1]	3	3	2	1	One free variable appears, so the map is not injective.
[1 0; 0 1; 1 1]	3	2	2	0	Full column rank gives an injective transformation into R³.
[1 2 0; 0 0 0]	2	3	1	2	Large nullity means many vectors map to the zero vector.

Formula used

Rank-nullity theorem: rank(A) + nullity(A) = n, where n is the number of columns in matrix A.

Left nullity: m - rank(A), where m is the number of rows.

Injective test: nullity(A) = 0.

Surjective test: rank(A) = m for a map from Rⁿ to R^m.

Consistency test for Ax = b: the system is consistent when rank(A) = rank([A|b]).

Unique solution test: a consistent system has a unique solution when rank(A) = n.

How to use this calculator

Enter the matrix with one row per line.
Separate values using spaces, commas, or semicolons.
Optionally enter vector b to analyze the system Ax = b.
Adjust the pivot tolerance if your matrix includes very small decimals.
Enable row reduction steps when you want instructional detail.
Click the calculate button to show results above the form.
Review rank, nullity, bases, RREF, and mapping properties.
Download the summary as CSV or print it as PDF.

Frequently asked questions

1. What does rank measure in a matrix?

Rank measures the number of linearly independent rows or columns. It tells you the dimension of the image, or output space, generated by the linear transformation.

2. What is nullity?

Nullity is the number of free variables in Ax = 0. It equals the dimension of the kernel, which contains all vectors sent to the zero vector.

3. Why is RREF useful here?

RREF makes pivot columns and free variables easy to identify. That lets you compute rank, nullity, basis vectors, and solution structure without guessing.

4. When is a transformation injective?

A linear transformation is injective when its kernel contains only the zero vector. In matrix terms, that means nullity equals zero and every column is a pivot column.

5. When is a transformation surjective?

For a map from Rⁿ to Rᵐ, surjectivity happens when rank equals m. Every output vector in the codomain can then be produced by some input vector.

6. What does rank([A|b]) tell me?

Comparing rank(A) and rank([A|b]) tells whether Ax = b is consistent. If the augmented rank is larger, the system has no solution.

7. Can this calculator test invertibility?

Yes. For square matrices, the calculator reports the determinant and whether full rank holds. A square matrix is invertible exactly when rank equals its size.

8. Which input separators are accepted?

You can separate numbers with spaces, commas, or semicolons. Put each matrix row on a new line, and enter the optional vector using the same separators.