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

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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.

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.

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.

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.

The rank of a matrix is equal to the rank of its transpose.

The rank of a null matrix is equal to 0, because it contains no non-zero vectors.

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