Rank of a Matrix

With each matrix, you can associate a non-negative integer, called its rank. The concept of rank plays an important role in solving a system of homogeneous and non-homogeneous equations.

A matrix obtained by leaving some rows and columns from the matrix A is called a submatrix of A. The determinant of any square submatrix of the given matrix A is called a minor of A. If the square submatrix is of order r, then the minor is also said to be of order r.

The matrix A is said to be of rank r, if

  1. A has at least one minor of order r which does not vanish.
  2. Every minor of A of order (r+1) and higher order vanishes.

The rank of a matrix is the order of any highest order non vanishing minor of the matrix.

The rank of A is denoted by the symbol ρ(A). The rank of a null matrix is defined to be zero.

The rank of the unit matrix of order n is n.

The rank of an m × n matrix A cannot exceed the minimum of m and n.

ρ(A) ≤ min {m, n}