Tool to calculate the rank of a Matrix. In mathematics, The rank of a matrix M is the number of linearly independent rows or columns.
Rank of a Matrix - dCode
Tag(s) : Matrix
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Rank of a Matrix
Matrix Rank Calculator
Dependant Vectors Calculator
Answers to Questions (FAQ)
What is the matrix rank? (Definition)
The rank of a matrix (sometimes noted as Rk) is mainly defined as the maximum number of row vectors (or column vectors) which are linearly independent.
The rank of a matrix is also the dimension of the vector subspace created by the vectors (either rows or columns) of the matrix.
The rank can be calculated for both rows and columns, it will be the same value.
How to calculate a matrix rank?
To calculate the rank of a $ M $ matrix, compare each of the rows between them and each of the columns between them to verify that they are two-by-two linearly independent.
Example: $$ M = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 2 & 2 & 4 \end{bmatrix} $$ The matrix $ M $ has rank $ 2 $ because row 2 is twice the row 1, they are not linearly independent.
NB: column 3 is the sum of columns 1 and 2, they are not linearly independent.
Why calculate the rank of a matrix?
The rank of a matrix provides information on several properties of the matrix, including:
— The number of solutions of the linear system associated with the matrix: if the rank of the matrix is equal to the number of variables, then the system admits a unique solution. If the rank is less than the number of variables, then the system admits an infinity of solutions.
— The invertibility of the matrix: a square matrix is invertible if and only if its rank is equal to its number of rows (or columns).
— The dimension of the space generated by the vectors of the matrix: the rank of a matrix is equal to the dimension of the vector space generated by its vectors.
What is the difference between the rank of a matrix and its order?
The order is the number of rows and columns it contains. The rank of a matrix provides information on its linear size, which cannot be larger than its order.
What is the rank of the transpose of a matrix?
The rank of a matrix is equal to the rank of its transpose.
What is the rank of a zero matrix?
The rank of a null matrix is equal to 0, because it contains no non-zero vectors.
Source code
dCode retains ownership of the "Rank of a Matrix" source code. Except explicit open source licence (indicated Creative Commons / free), the "Rank of a Matrix" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Rank of a Matrix" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Rank of a Matrix" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cita dCode
The copy-paste of the page "Rank of a Matrix" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Rank of a Matrix on dCode.fr [online website], retrieved on 2024-05-07,
