The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it represents the maximum number of independent rows or columns that can collectively span the entire row space or column space of the matrix.

  To determine the rank of a matrix, one can use various methods such as row reduction or computing the determinant of certain submatrices. The most commonly used method is Gaussian elimination, which transforms the matrix into row-echelon form or reduced row-echelon form.

  The rank of a matrix has several important properties:

  1. The rank of a matrix is always less than or equal to the minimum of the number of rows and columns in the matrix.

  2. The rank of a matrix is equal to the number of nonzero rows in its row-echelon form or reduced row-echelon form.

  3. The rank of a matrix is equal to the number of nonzero singular values in its singular value decomposition.

  The rank of a matrix plays a crucial role in various applications such as solving systems of linear equations, computing eigenvalues and eigenvectors, and determining the dimension of the column and null space of a matrix. Additionally, the rank-nullity theorem states that the rank of a matrix plus the nullity (dimension of the null space) is equal to the number of columns in the matrix.

  In summary, the rank of a matrix provides important information about the linearly independent components of the matrix and is used in various mathematical computations and applications.