Matrix Rank Properties Calculator

Calculator Form

Use spaces or commas between values. Separate rows with new lines. The form uses 3 columns on large screens, 2 on medium screens, and 1 on mobile.

Example Data Table

Example Matrix	Dimensions	Rank	Nullity	Observation
[1 2 3; 2 4 6; 1 1 1]	3 × 3	2	1	One row depends on another, so the matrix is singular.
[1 0; 0 1]	2 × 2	2	0	Identity matrices have full rank and are invertible.
[1 2 3; 2 4 6]	2 × 3	1	2	Rows are multiples, so only one pivot exists.

Formula Used

Primary definitions

Rank(A) is the number of pivot positions in the reduced row echelon form of matrix A.

Nullity(A) = n − rank(A), where n is the number of columns of A.

Full rank means the rank reaches the largest possible value, min(m, n).

For square matrices, nonzero determinant implies full rank and invertibility.

Rank property checks

rank(A) = rank(A^T)

rank(A + B) ≤ rank(A) + rank(B)

rank(AB) ≤ min(rank(A), rank(B))

rank(AB) ≥ rank(A) + rank(B) − n for compatible matrices using Sylvester’s inequality.

How to Use This Calculator

Enter the row count and column count for Matrix A.

Type the matrix entries using spaces or commas between values.

Place each row on a new line so the parser reads them correctly.

Adjust the pivot tolerance only if you need stricter numerical behavior.

Enable Matrix B when you want addition and multiplication rank checks.

Press the submit button to show the result above the form.

Review the RREF, pivot columns, inequalities, graphs, and summary tables.

Use the CSV or PDF buttons to export the computed output.

FAQs

1) What does matrix rank measure?

Matrix rank measures how many rows or columns are linearly independent. It also tells you the dimension of both the row space and column space.

2) Why does the calculator show nullity?

Nullity shows how many free variables appear in a homogeneous system. It is found by subtracting the rank from the number of columns.

3) Why is RREF useful for rank?

Reduced row echelon form exposes pivot positions clearly. Counting those pivots gives the rank directly and also helps identify dependent rows or columns.

4) Does rank change after transposing a matrix?

No. A matrix and its transpose always have the same rank. The calculator verifies this property automatically for Matrix A.

5) When is a square matrix invertible?

A square matrix is invertible exactly when its rank equals its size. In that case, its determinant is nonzero and its nullity is zero.

6) Why can rank(A + B) be smaller than rank(A) + rank(B)?

Addition can cause cancellation between row or column structures. Because of that, the rank of a sum often stays below the simple total.

7) What does the AB rank check tell me?

It checks that multiplying matrices cannot create more independent directions than the factors already contain. This helps confirm major product-rank inequalities.

8) What matrix sizes work best here?

The page allows sizes up to 8 × 8 per matrix for clarity and speed. Smaller inputs also keep the tables, heatmaps, and exports easier to read.