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rank


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(of a matrix)

Let A be an m×n matrix. The column rank of A is the largest number of elements in a linearly independent set of columns of A. The row rank of A is the largest number of elements in a linearly independent set of rows of A. It can be shown that elementary row operations on a matrix do not change the column rank or the row rank. Consequently, the column rank and row rank of A are equal, both being equal to the number of non-zero rows in the matrix in reduced echelon form to which A can be transformed. This common value is the rank of A. It can also be shown that the rank of A is equal to the number of rows and columns in the largest square submatrix of A that has non-zero determinant. An n×n matrix is invertible if and only if it has rank n.

Subjects: Mathematics.


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