Matrix Rank

The matrix rank calculator finds the rank of a rectangular matrix up to 6×6 by Gaussian elimination: it reduces the matrix to row echelon form and counts the non-zero rows. All row operations and the pivot columns are shown.

What is the rank

The rank of a matrix is the largest number of linearly independent rows (or columns). It is also the largest order of a non-zero minor. The rank does not exceed the smaller of the numbers of rows and columns.

How to find the rank by Gaussian elimination

1. Use row operations (swapping, multiplying by a non-zero number, adding a multiple of another row) to reach row echelon form.
2. These operations do not change the rank.
3. The rank is the number of non-zero rows in the echelon form.

Example. A = (1 2 3 4; 2 4 6 8; 3 1 2 5).
The second row is twice the first and becomes zero; two non-zero rows remain — rank A = 2.

Where the rank is used

• Rouché–Capelli theorem: a linear system is consistent if the rank of its matrix equals the rank of the augmented matrix.
• An n×n matrix is invertible if its rank is n.
• The rank is the dimension of the space spanned by the rows (columns).