Finding the rank of a matrix: practice with solutions
The rank is the number of linearly independent rows. The easiest way to find it is Gaussian elimination: reach row echelon form and count the rows that are not all zeros.
How to do it
- 1
Bring the matrix to row echelon form with row operations.
- 2
Rows that contain only zeros were combinations of the other rows.
- 3
The rank is the number of remaining rows, i.e. the number of pivots.
- 4
Check: the rank is at most the smaller of the number of rows and columns.
Example with worked solution
Worked solution
Starting matrix
−1 −1 −2 2 2 5 −1 −1 −1 1. II + 2·I
−1 −1 −2 0 0 1 −1 −1 −1 2. III − I
−1 −1 −2 0 0 1 0 0 1 3. III − II
−1 −1 −2 0 0 1 0 0 0
Echelon form: 2 rows are not all zeros, so rank = 2.
Your turn
Enter the row operations, e.g. II − 3·I. If you get stuck, "Hint: next step" helps.
Find the rank of the matrix.
Bring the matrix to row echelon form and count the rows that are not all zeros.
| −1 | 1 | 1 |
| −1 | 1 | −1 |
| 2 | −2 | 4 |
Your answer
Typical mistakes
- Stopping before echelon form: two rows can still be dependent even though no zero row is visible yet.
- Counting rows that contain some zero: only rows made entirely of zeros matter, and those drop out.
- Giving rank 4 for a wide matrix (e.g. 3×4): the rank can never exceed the number of rows.
Frequently asked questions
What does the rank say about a linear system?
If rank(A) = rank(A|b), the system is solvable. If that rank also equals the number of unknowns, there is exactly one solution (Rouché–Capelli theorem).
Row rank or column rank?
They are always equal, so it is enough to bring the rows to echelon form.
When does a square matrix have full rank?
Exactly when its determinant is not 0. It is then invertible as well.