The rank of a matrix is a very important concept in the use of matrices to solve a set of linear equations. The rank of a matrix is also used to identify the linear independence of the rows (or columns) of a matrix.  An n x m  matrix A,  has a rank of r if it contains at least one r - rowed  square submatrix with non vanishing determinant. Also, the determinant of any r + 1 or more square submatrix of A is zero.

Some useful information:

The rank of an n x m  matrix can at most have rank equal to the minimum of the two dimensions m or n

An n x n matrix A has a rank r < n only if |A| = 0

If |A| = 0 then A is called a singular matrix. Otherwise it is non singular

An n x n square matrix has rank r = n only if |A| ~= 0

If rank of A = 0 then A is a null  matrix

The rank of the transpose of a matrix is the same as the original matrix

The rank of the matrix A is equal to the maximum number of linearly independent rows (or columns) of A