Chapter Rank of Matrix All Lectures are available on this page  Rank of Matrix :  Definition: A number r r  is said to be the rank of a matrix A A  if it possesses the following two properties: (i) There is at least one square submatrix of A A  of order r r  whose determinant is not equal to zero. (ii) If the matrix A A  contains any square submatrix of order r + 1 r + 1 , then the determinant of every square submatrix of A A  of order r + 1 r + 1  should be zero. In short, the rank of a matrix is the order of any highest order non-vanishing minor of the matrix. Thus, the rank of a matrix A A  is the order of any highest order square submatrix of A A  whose determinant is not equal to zero. We shall denote the rank of a matrix A by the symbol ρ(A). It is obvious that the rank r r  of an ( m × n ) (m \times n)  matrix can at most be equal to the smaller of the numbers m m  and n n , but it may be less. If...

Chapter Matrices All Lectures are available on this page  Matrices : A set of mn numbers (real or complex) arranged in the form of a rectangular array having m rows and n columns is called an m x n matrix [to be read as ‘m by n’ matrix]. An m x n matrix is usually written as: A = [ a 11 a 12 … … a 1 n a 21 a 22 … … a 2 n a 31 a 32 … … a 3 n ⋮ ⋮ ⋱ ⋮ ⋮ a m 1 a m 2 … … a m n ] A = \begin{bmatrix} a_{11} & a_{12} & \dots & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & \dots & a_{2n} \\ a_{31} & a_{32} & \dots & \dots & a_{3n} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \dots & \dots & a_{mn} \end{bmatrix} ​ In a compact form, the above matrix is represented by: A = [ a i j ] i = 1 , 2 , 3 , … , m , j = 1 , 2 , 3 , … , n or simply by [ a i j ] m × n​ We write the general element of the matrix and enclose it in brackets of type [ ] or of the type ( ) . (i) Square Matrices Definitio...