BYTETOOLS

Matrix Rank Calculator

Find the rank of any matrix by exact row reduction, then get the nullity from the rank-nullity theorem plus full row rank, column rank and singularity.

2
Rank
1
Nullity
3×3
Size
1, 2
Pivot columns

Row echelon form used to count the rank

10-1
012
000

Pivot columns are highlighted. Rank = number of pivots = number of non-zero rows after reduction.

What the rank tells you

  • Rank = 2. Row reduction left 2 non-zero rows, and the rank is the count of those rows.
  • Rank-nullity: rank + nullity = number of columns, so 2 + 1 = 3. The null space of this matrix has dimension 1.
  • The matrix is rank deficient — neither the rows nor the columns are linearly independent.
  • Square but rank 2 < 3, so the matrix is singular: its determinant is 0 and it has no inverse.

What is the Matrix Rank Calculator?

The ByteTools Matrix Rank Calculator row-reduces your matrix to echelon form and counts the non-zero rows — that count is the rank, the number of genuinely independent rows or columns your matrix has.

  • Rank from exact rational row reduction — no floating-point false zeros
  • Nullity computed with the rank-nullity theorem and shown alongside
  • Pivot column positions listed
  • Full row rank, full column rank and singular or invertible verdicts in plain English
  • Handles rectangular matrices up to 10×10, with fractions accepted
  • Runs 100% in your browser — nothing uploaded

How to use the Matrix Rank Calculator

  1. 1

    Enter your matrix with one row per line and values separated by spaces or commas.

  2. 2

    Read the rank and nullity from the summary tiles at the top.

  3. 3

    Check the echelon form below, where the pivot columns are highlighted.

  4. 4

    Read the notes to see whether the matrix has full row or column rank, and whether it is invertible.

  5. 5

    Copy the reduced matrix or the notes with the copy buttons.

About the Matrix Rank Calculator

The ByteTools Matrix Rank Calculator row-reduces your matrix to echelon form and counts the non-zero rows — that count is the rank, the number of genuinely independent rows or columns your matrix has. Because the reduction is done in exact fractions rather than floating point, a value is either exactly zero or it is not, so the rank is never wrong because of rounding noise.

Alongside the rank you get the nullity from the rank-nullity theorem (rank + nullity = number of columns), the list of pivot column positions, and plain-English verdicts: whether the matrix has full row rank, full column rank, both, or neither. For a square matrix it also tells you whether the matrix is invertible or singular, since full rank and a non-zero determinant are the same condition.

Everything is computed by JavaScript running locally in your browser. Your matrix is never uploaded, no data is stored, and the page still works when you are offline.

Frequently asked questions

What is the rank of a matrix?

The rank is the number of linearly independent rows, which is always equal to the number of linearly independent columns. Practically, it is the count of non-zero rows left after you row-reduce the matrix — the amount of genuinely distinct information it carries.

What is the rank-nullity theorem?

It states that for any matrix, rank + nullity = the number of columns. So if a 3×5 matrix has rank 2, its null space has dimension 3. This calculator shows both numbers and the arithmetic connecting them.

What does full rank mean?

Full rank means the rank equals the smaller of the row count and column count. Full row rank means the rows are independent; full column rank means the columns are, and that the null space is trivial. A square matrix with full rank is invertible.

How do you know if a matrix is singular from its rank?

A square n×n matrix is singular exactly when its rank is less than n. That is equivalent to a zero determinant and to having no inverse. This tool states which case you are in whenever your input is square.

Can the rank of a matrix be zero?

Only for the zero matrix, where every entry is 0. Any matrix with at least one non-zero entry has rank of at least 1, because that row on its own is linearly independent.

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