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Rank of a Matrix

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.

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Rank of a Matrix -

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# Rank of a Matrix

## Matrix Rank Calculator

### Modular Rank of a Matrix

The modulus is only applicable to integer matrices, with N a prime number

## Dependant Vectors Calculator

 Search Relations between columns C Relations between rows R

### 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.

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