The rank of a matrix is said to be r, if
(i) It has atleast minors of order r is different from zero.
(ii) All minors of A of order higher than r are zero.
The rank of A is denoted by `rho(A).`

Note 1 : The rank of a zero matrix is zero and the rank of an identity matrix of order `n` is `n.`
Note 2 : The rank of a matrix in echelon form is equal to the number of non - zero rows of the matrix.
Note 3 : The rank of a non-singular matrix ( !AI ;e 0) of order n is n.

`text(Properties of Rank of Matrices :)`

(i) If `A= [a_(ij)]_(m xx n)` and `B= [b_(ij)]_(mxxn)` , then

`rho (A +B) <= rho(A) + rho(B) `

(ii) `A= [a_(ij)]_(m xx n)` and `B= [b_(ij)]_(nxxp)` , then

`rho(AB) <= rho(A)` and `rho(AB) <= rho(B)`

(iii) If `A= [a_(ij)]_(nxxn) ,` then `rho(A) = rho(A')`