Matrix Rank Calculator
Determine the rank of a matrix by row-reducing to echelon form.
Inputs
Rank
2
Reduced Row Echelon Form
1 0 -1 0 1 2 0 0 0
Pivot Columns
1, 2
Nullity
1
Step by step
Original Matrix
= 1 2 3 4 5 6 7 8 9
RREF
= 1 0 -1 0 1 2 0 0 0
Pivot columns (1-indexed)
= 1, 2
Rank
= 2
Nullity (cols − rank)
= 1
How it works
The rank of a matrix is the number of linearly independent rows (or equivalently columns). It equals the number of pivot positions in the reduced row echelon form. The rank-nullity theorem states: rank + nullity = number of columns.
Formula
Rank-Nullity Theorem
rank(A) + nullity(A) = n (number of columns)
- A
- Matrix
- n
- Number of columns
Frequently Asked Questions
What does rank tell us about a system of equations?
If rank equals the number of unknowns, the system has a unique solution. If rank is less, the system has infinitely many solutions or is inconsistent.
Can the rank exceed the number of rows or columns?
No. The rank is at most min(m, n) for an m×n matrix.
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