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Rank of a Matrix

Matrix rank calculator. Enter a matrix to compute rank(A) by row reduction—for rank and system classification.

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rank(A) = number of nonzero rows
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Rank of a Matrix

1. Definition
After row-reducing A to echelon form, the number of nonzero rows is its rank, rank(A).
2. Symbols
SymbolMeaning
rank(A)rank of A
m, nrows and columns
full rankrank = min(m,n)
3. How this tool works
  • Elementary row operations preserve rank;
  • Rank equals the size of a basis of the row space;
  • A square matrix is invertible iff it is full rank.
4. Steps
  1. Enter A;
  2. Click Calculate to see the rank;
  3. Compare with min(m,n) to check full rank.
5. Example
Example: A=1 2 3;4 5 6;7 8 9. Row 3 = 2·row2 − row1, so echelon form has 2 nonzero rows, rank=2.
6. Pitfalls
Rank never exceeds min(m,n);
The zero matrix has rank 0;
Only full-rank square matrices are invertible.

❓ FAQ｜Rank of a Matrix

How is rank defined?
The number of nonzero rows in row echelon form.
Maximum rank?
Never more than min(m,n).
Why does full rank matter?
A square matrix is full rank iff it is invertible iff det≠0.
Rank of the zero matrix?
0.
Do row/column operations change rank?
No.