Matrix Rank
Rank is the number of linearly independent rows (or columns). It equals the number of pivot positions in RREF. rank(A) + nullity(A) = n (columns). Full rank โบ invertible for square matrices.
Why This Mathematical Concept Matters
Why: Rank determines solvability of Ax = b, dimension of column/row space, and invertibility. Central to linear algebra.
How: Reduce to RREF via Gaussian elimination. Count pivot columns (or rows). rank = number of pivots.
- โrank(A) = rank(Aแต).
- โrank(AB) โค min(rank(A), rank(B)).
- โFull rank โบ columns linearly independent.
Examples
Rank vs Nullity
Rank / Nullity
Steps
โ ๏ธFor educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
rank = dim(column space)
rank = dim(row space)
Key Takeaways
- โข Rank = number of linearly independent rows (or columns). For mรn matrix, rank โค min(m,n).
- โข Gaussian elimination reduces to row echelon form; count non-zero pivot rows.
- โข Nullity = n โ rank (columns โ rank). Rank-nullity: rank + nullity = n.
- โข Full rank: rank = min(m,n). Square matrix is invertible iff rank = n.
- โข Zero matrix has rank 0; identity has rank n.
Did You Know?
How It Works
1. Gaussian elimination
Apply row operations to reduce matrix to row echelon form (REF).
2. Pivot positions
Leading non-zero entry in each row. Count them to get rank.
3. RREF
Further reduce to reduced row echelon form (1s on pivots, zeros above/below).
4. Nullity
nullity = cols โ rank. Dimension of null space (solutions to Ax = 0).
Expert Tips
Choose pivots wisely
Partial pivoting improves numerical stability.
Check linear dependence
If rank < min(m,n), rows/cols are linearly dependent.
System Ax=b
Consistent iff rank(A) = rank([A|b]). Unique solution iff full rank.
SVD alternative
For large matrices, SVD gives rank via singular values.
Comparison Table
| Feature | This Calculator | NumPy |
|---|---|---|
| RREF steps | โ | โ |
| Pivot positions | โ | โ |
| Nullity | โ | โ |
| Rectangular matrices | โ | โ |
FAQ
What is rank?
The maximum number of linearly independent rows (or columns). Equals dimension of column/row space.
What is nullity?
Dimension of null space: nullity = n โ rank. Solutions to Ax = 0.
When is a matrix full rank?
When rank = min(rows, cols). Square matrix full rank iff invertible.
How does Gaussian elimination work?
Row operations: swap, scale, add multiple. Pivots become 1, entries below become 0.
What is RREF?
Reduced row echelon form: leading 1s, zeros above and below each pivot.
Can rectangular matrices have full rank?
Yes. 2ร4 matrix can have rank 2 (max for 2 rows).
What is rank of zero matrix?
Zero. No linearly independent rows.
Relation to eigenvalues?
For square A, rank = number of non-zero eigenvalues (counting multiplicity).
Stats
Sources
- โข Gilbert Strang, Linear Algebra and Its Applications
- โข Khan Academy โ Linear Algebra
- โข MIT OpenCourseWare 18.06
- โข Wolfram MathWorld โ Matrix Rank
- โข 3Blue1Brown โ Essence of Linear Algebra
- โข Golub & Van Loan, Matrix Computations