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Linear Algebra

Matrix Rank

The rank of a matrix is the number of linearly independent rows — equivalently, the number of pivots in its row echelon form. Enter the matrix and the calculator row-reduces it and reports the rank.

Matrix Rank

Number of linearly independent rows — by row reduction.

Try:
Answerrank(A) = 2
  1. Matrix A3×3 [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
  2. MethodRow reduce to echelon form; the rank is the number of non-zero rows (= number of pivots).
  3. Row operationsSwap R1 ↔ R3; R1 → (1/7)·R1; R2 → R2 − (4)·R1; R3 → R3 − (1)·R1; Swap R2 ↔ R3; R2 → (1/0.857143)·R2; R1 → R1 − (1.14286)·R2; R3 → R3 − (0.428571)·R2
  4. Reduced form[[1, 0, -1], [0, 1, 2], [0, 0, 0]]
  5. Pivot columnscolumn 1, column 2
  6. Rankrank(A) = 2
  7. NoteRank < min(rows, cols) = 3: the rows are linearly dependent.
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Frequently asked questions

Is row rank the same as column rank?

Yes. For every matrix the number of linearly independent rows equals the number of linearly independent columns — this common value is the rank.

What does full rank mean?

An m×n matrix has full rank when its rank equals min(m, n). A square matrix has full rank exactly when it is invertible.

How does row reduction reveal the rank?

After reducing to row echelon form, each leading 1 corresponds to one independent row. The number of non-zero rows is the rank.

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