The rank of a matrix is a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by the matrix). In linear algebra, the rank of a matrix A is the dimension of the vector space generated by its columns, which corresponds to the maximal number of linearly independent columns of A). This is identical to the dimension of the vector space spanned by its rows). The rank of a matrix is commonly denoted by rank(A) or rk(A), and it is one of its most fundamental characteristics).

There are multiple equivalent definitions of rank, including the column rank and the row rank of a matrix). The column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A). The rank of A is the maximal number of linearly independent columns of A, which is the dimension of the column space of A). Similarly, the rank of A is the maximal number of linearly independent rows of A, which is the dimension of the row space of A).

To find the rank of a matrix, one can transform the matrix into its echelon form and count the number of non-zero rows. The rank of a matrix is the number of linearly independent rows or columns in it. The rank of a null matrix is zero, as it has no non-zero rows or columns. The rank of a matrix cannot exceed the number of its rows or columns.