Matrix Calculator

1. Definition and formula: What is the rank of a matrix?

The rank of a matrix is the number of linearly independent rows or columns. Equivalently, the rank is the number of pivot positions in the matrix's reduced row echelon form (RREF).

Rank applies to both square and rectangular matrices. Row rank equals column rank.

Practical method (summary):

1. Compute the RREF of matrix $A$ (Gauss–Jordan elimination).
2. Count the number of nonzero (pivot) rows in RREF.
3. That count is $\operatorname{rank}(A)$.

2. How to compute rank (quick recipe)

• Convert $A$ to RREF using elementary row operations.
• Count pivot rows (rows with leading 1's).
• Alternatively, count maximal number of linearly independent columns (column rank).

3. Worked examples

Example 1

Compute the rank of

Row-reduce (summary):

There are 2 pivot rows, so

Example 2

Find the rank of

Row-reduce (summary):

Two pivot rows ⇒

4. Common mistakes (pitfalls)

• Counting nonzero entries: Rank counts independent rows/columns, not nonzero entries.
• Stopping too early in elimination: REF might hide dependencies; RREF clarifies pivots.
• Mixing determinant logic: For square matrices, $\det\neq 0$ ⇒ full rank, but determinant is not defined for non-square matrices.
• Assuming visually independent: Rows that look different may still be linearly dependent—use elimination to be sure.

5. Practice problems (click to show answers)

Try these. Answers are hidden — click Show Answer to reveal each rank.

Exercise 1

Exercise 2

Exercise 3

Exercise 4

To study matrix rank from different angles, try the matrix inverse calculator, matrix multiplication calculator, and LU decomposition calculator.