MatrixRank
✖
MatrixRank
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Details and Options
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- MatrixRank works on both numerical and symbolic matrices.
- The rank of a matrix is the number of linearly independent rows or columns.
- MatrixRank[m,Modulus->n] finds the rank for integer matrices modulo n.
- MatrixRank[m,ZeroTest->test] evaluates test[m[[i,j]]] to determine whether matrix elements are zero. The default setting is ZeroTest->Automatic.
- MatrixRank[m,Tolerance->t] gives the minimum rank with each element in a numerical matrix assumed to be correct only to within tolerance t.
- MatrixRank works with sparse arrays and structured arrays.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Find the number of linearly independent rows of a numerical matrix:
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https://wolfram.com/xid/0cg3r9da2-3io3t
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Find the number of linearly independent rows of a symbolic matrix:
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https://wolfram.com/xid/0cg3r9da2-tjk2jw
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Compute the rank of a rectangular matrix:
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https://wolfram.com/xid/0cg3r9da2-3foezn
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Scope (12)Survey of the scope of standard use cases
Basic Uses (7)
Find the rank of a machine-precision matrix:
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https://wolfram.com/xid/0cg3r9da2-gzis3e
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https://wolfram.com/xid/0cg3r9da2-bjja1q
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Rank of an exact, rectangular matrix:
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https://wolfram.com/xid/0cg3r9da2-zo4y0v
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https://wolfram.com/xid/0cg3r9da2-piv3b5
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Find the rank of an arbitrary-precision matrix with more rows than columns:
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https://wolfram.com/xid/0cg3r9da2-yo7vbd
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https://wolfram.com/xid/0cg3r9da2-p8sjhl
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Compute the rank of a symbolic matrix:
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https://wolfram.com/xid/0cg3r9da2-fxlyis
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MatrixRank assumes all symbols to be independent:
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https://wolfram.com/xid/0cg3r9da2-gtn59g
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Computing the rank of large machine-precision matrices is efficient:
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https://wolfram.com/xid/0cg3r9da2-dkq7nk
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https://wolfram.com/xid/0cg3r9da2-lx8juz
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Compute the rank of a matrix with finite field elements:
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https://wolfram.com/xid/0cg3r9da2-nw2qll
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Special Matrices (5)
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https://wolfram.com/xid/0cg3r9da2-ocj3kf
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https://wolfram.com/xid/0cg3r9da2-fm3xwv
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The rank of structured matrices:
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https://wolfram.com/xid/0cg3r9da2-ebpduc
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https://wolfram.com/xid/0cg3r9da2-dhkxvx
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https://wolfram.com/xid/0cg3r9da2-wdjmwi
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https://wolfram.com/xid/0cg3r9da2-j45yx1
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IdentityMatrix always has full rank:
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https://wolfram.com/xid/0cg3r9da2-8vb729
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https://wolfram.com/xid/0cg3r9da2-pflv9c
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HilbertMatrix always has full rank:
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https://wolfram.com/xid/0cg3r9da2-q1l839
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Compute the rank of a matrix of univariate polynomials of degree
:
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https://wolfram.com/xid/0cg3r9da2-cupbp5
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https://wolfram.com/xid/0cg3r9da2-j0guy7
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Options (2)Common values & functionality for each option
Modulus (1)
The rank of a matrix depends on the modulus used:
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https://wolfram.com/xid/0cg3r9da2-eozjbj
With ordinary arithmetic, m has the full rank of 3:
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https://wolfram.com/xid/0cg3r9da2-b00497
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With arithmetic modulo 5, the rank is only 2:
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https://wolfram.com/xid/0cg3r9da2-db5dy3
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Tolerance (1)
The setting of Tolerance can affect the estimated rank for numerical ill-conditioned matrices:
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https://wolfram.com/xid/0cg3r9da2-b22du7
In exact arithmetic, m has full rank:
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https://wolfram.com/xid/0cg3r9da2-cs7883
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With machine arithmetic, the default is to consider elements that are too small as zero:
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https://wolfram.com/xid/0cg3r9da2-d2wixt
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With zero tolerance, even small terms may be taken into account:
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https://wolfram.com/xid/0cg3r9da2-nxde1
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With a tolerance greater than the pivot in the middle row, the last two rows are considered zero:
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https://wolfram.com/xid/0cg3r9da2-nkdcs6
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Applications (11)Sample problems that can be solved with this function
Spans and Linear Independence (5)
The following three vectors are not linearly independent:
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https://wolfram.com/xid/0cg3r9da2-gocnsl
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Therefore the matrix rank of the matrix whose rows are the vectors is 2:
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https://wolfram.com/xid/0cg3r9da2-l8nbp1
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The following three vectors are linearly independent:
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https://wolfram.com/xid/0cg3r9da2-keoxxm
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Therefore the matrix rank of the matrix whose rows are the vectors is 3:
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https://wolfram.com/xid/0cg3r9da2-8mo3ba
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Determine if the following vectors are linearly independent or not:
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https://wolfram.com/xid/0cg3r9da2-qnovlz
The rank of the matrix formed from the vectors is less than four, so they are not linearly independent:
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https://wolfram.com/xid/0cg3r9da2-sotoh0
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Find the dimension of the column space of the following matrix:
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https://wolfram.com/xid/0cg3r9da2-fq6172
The dimension of the space of all linear combinations of the columns equals the matrix rank:
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https://wolfram.com/xid/0cg3r9da2-oi7hun
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Find the dimension of the subspace spanned by the following vectors:
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https://wolfram.com/xid/0cg3r9da2-shxd3q
Since the matrix rank of the matrix formed by the vectors is three, that is the dimension of the subspace:
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https://wolfram.com/xid/0cg3r9da2-doh8f0
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Equation Solving and Invertibility (6)
Determine if the following system of equations has a unique solution:
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https://wolfram.com/xid/0cg3r9da2-upikkb
Rewrite the system in matrix form:
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https://wolfram.com/xid/0cg3r9da2-zs7ify
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The coefficient matrix has full rank, so the system has a unique solution:
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https://wolfram.com/xid/0cg3r9da2-6x59zk
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Verify the result using Solve:
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https://wolfram.com/xid/0cg3r9da2-tqujfz
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Determine if the following matrix has an inverse:
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https://wolfram.com/xid/0cg3r9da2-soeczx
The rank is less than the dimension of the matrix, so it is not invertible:
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https://wolfram.com/xid/0cg3r9da2-1itznp
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Verify the result using Inverse:
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https://wolfram.com/xid/0cg3r9da2-sfsjp9
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Determine if the following matrix has a nonzero determinant:
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https://wolfram.com/xid/0cg3r9da2-1q7xab
Since it has full rank, its determinant must be nonzero:
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https://wolfram.com/xid/0cg3r9da2-0wktjf
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Confirm the result using Det:
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https://wolfram.com/xid/0cg3r9da2-xxnsh
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is an eigenvalue of
if
does not have full rank. Moreover, a matrix is deficient if it has an eigenvalue whose multiplicity is greater than the difference between the rank of
and the number of columns. Show that
is an eigenvalue for the following matrix
:

https://wolfram.com/xid/0cg3r9da2-5fmzho

https://wolfram.com/xid/0cg3r9da2-qf2h5t

Confirm the result using Eigenvalues:

https://wolfram.com/xid/0cg3r9da2-x53b0b
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The matrix is deficient because 2 appears twice, but the difference in rank is only one:

https://wolfram.com/xid/0cg3r9da2-dxfyw9

Confirm the result with Eigensystem, which indicates deficiency by padding the eigenvector list with zeros:

https://wolfram.com/xid/0cg3r9da2-qbcxdp
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Most but not all random 10×10 0–1 matrices have full rank:

https://wolfram.com/xid/0cg3r9da2-by0vsd
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Estimate the average rank of a random 10×10 0–1 matrix:

https://wolfram.com/xid/0cg3r9da2-hlp98e

Find the ranks of coprimality arrays:

https://wolfram.com/xid/0cg3r9da2-t8utl
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https://wolfram.com/xid/0cg3r9da2-jac2l
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
https://wolfram.com/xid/0cg3r9da2-joxgvx
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Compute the first 50 such arrays. Only the first three have full rank:

https://wolfram.com/xid/0cg3r9da2-gawh6p

Visualize the growth in rank versus the dimension of the matrix:

https://wolfram.com/xid/0cg3r9da2-fj1rnx
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Properties & Relations (9)Properties of the function, and connections to other functions
By the rank-nullity theorem, MatrixRank[m] is the number of columns minus the dimension of the null space:

https://wolfram.com/xid/0cg3r9da2-gxkm9m
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https://wolfram.com/xid/0cg3r9da2-f42s8h
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https://wolfram.com/xid/0cg3r9da2-c0wcxx
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
https://wolfram.com/xid/0cg3r9da2-dzhaqc

The column and row rank of a matrix are equal:

https://wolfram.com/xid/0cg3r9da2-bh2j57

https://wolfram.com/xid/0cg3r9da2-d8v2ib

MatrixRank[m] equals the number of nonzero rows in RowReduce[m]:

https://wolfram.com/xid/0cg3r9da2-r3i6g7
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https://wolfram.com/xid/0cg3r9da2-tqbxyq
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
https://wolfram.com/xid/0cg3r9da2-ji740x

For a square matrix, m has full rank if and only if Det[m]!=0:

https://wolfram.com/xid/0cg3r9da2-wdl5f0

https://wolfram.com/xid/0cg3r9da2-m3c6cz
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https://wolfram.com/xid/0cg3r9da2-jdbcx4
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For a square matrix, m has full rank if and only if the null space is empty:

https://wolfram.com/xid/0cg3r9da2-3raccb

https://wolfram.com/xid/0cg3r9da2-5pyzjz


https://wolfram.com/xid/0cg3r9da2-fy748r

For a square matrix, m has full rank if and only if m has an inverse:

https://wolfram.com/xid/0cg3r9da2-xhqrgz

https://wolfram.com/xid/0cg3r9da2-5gvptr


https://wolfram.com/xid/0cg3r9da2-4o8e3f
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
For a square matrix, m has full rank iff LinearSolve[m,b] has a solution for a generic b:

https://wolfram.com/xid/0cg3r9da2-sduw6z

https://wolfram.com/xid/0cg3r9da2-oclh3t


https://wolfram.com/xid/0cg3r9da2-7noouu

MatrixRank[m] is equal to Length[SingularValueList[m]]:

https://wolfram.com/xid/0cg3r9da2-cm9elg

https://wolfram.com/xid/0cg3r9da2-h0mjv9
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
https://wolfram.com/xid/0cg3r9da2-bzanbk

The outer product of vectors has matrix rank 1:

https://wolfram.com/xid/0cg3r9da2-b0jbqq

https://wolfram.com/xid/0cg3r9da2-dzw1jd

Possible Issues (2)Common pitfalls and unexpected behavior
MatrixRank may depend on the precision of the given matrix:

https://wolfram.com/xid/0cg3r9da2-d63v26
Use exact arithmetic to compute the matrix rank exactly:

https://wolfram.com/xid/0cg3r9da2-x9m0c

Use machine arithmetic. Machine numbers cannot distinguish between and
:

https://wolfram.com/xid/0cg3r9da2-epi0zb
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Use 24‐digit-precision arithmetic:

https://wolfram.com/xid/0cg3r9da2-cnd3yv

MatrixRank assumes all symbols to be independent:

https://wolfram.com/xid/0cg3r9da2-hdj0vs

The special case gives a different result:

https://wolfram.com/xid/0cg3r9da2-nqgyjw

Wolfram Research (2003), MatrixRank, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixRank.html (updated 2024).
Text
Wolfram Research (2003), MatrixRank, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixRank.html (updated 2024).
Wolfram Research (2003), MatrixRank, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixRank.html (updated 2024).
CMS
Wolfram Language. 2003. "MatrixRank." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/MatrixRank.html.
Wolfram Language. 2003. "MatrixRank." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/MatrixRank.html.
APA
Wolfram Language. (2003). MatrixRank. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixRank.html
Wolfram Language. (2003). MatrixRank. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixRank.html
BibTeX
@misc{reference.wolfram_2025_matrixrank, author="Wolfram Research", title="{MatrixRank}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/MatrixRank.html}", note=[Accessed: 19-February-2025
]}
BibLaTeX
@online{reference.wolfram_2025_matrixrank, organization={Wolfram Research}, title={MatrixRank}, year={2024}, url={https://reference.wolfram.com/language/ref/MatrixRank.html}, note=[Accessed: 19-February-2025
]}