. . Rank of a Matrix To define the rank of a matrix, we have to know about sub-matrices and minors of a matrix. Let A be a given matrix.
A matrix obtained by deleting some rows and some columns of A is called a sub-matrix of A . A matrix is a sub-matrix of itself because it is obtained by leaving zero number of rows and zero number of columns. Recall that the determinant of a square sub-matrix of a matrix is called a minor of the matrix. Definition .
The rank of a matrix A is defined as the order of a highest order non-vanishing minor of the matrix A . It is denoted by the symbol ρ ( ). A The rank of a zero matrix is defined to be . - - Note (i) If a matrix contains at-least one non-zero element, then ρ ( ) A ≥ (ii) The rank of the identity matrix I n is n .
(iii) If the rank of a matrix A is r , then there exists at-least one minor of A of order r which does not vanish and every minor of A of order r + and higher order (if any) vanishes. (iv) If A is an m n matrix, then ρ ( ) , . m n m n ≤ min{ } = minimum of (v) A square matrix A of order n has inverse if and only if ρ ( ) Example . Find the rank of each of the following matrices: (i) (ii) (i) Let A = .
Then A is a matrix of order × . So ρ ( ) min A ≤