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amos ppt srms.pptx

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amos ppt srms.pptx

  1. 1. Rank (By using normal & echelon form) Amos Richard
  2. 2. Rank • Rank is denoted by (‘r’)( ‘ρ’), Rank of a matrix of independent rows or colums. • Only null matrix or zero matrix has zero rank. • For a matrix of order rank (ρ(A)) od ‘A’ is less than or equal to minimum (m,n). [A]mxn ρ(A) ≤ min(m,n). • For a non-singular matrix ‘A’ of order ‘n’ ρ(A)=n
  3. 3. Echelon Form Method For Rank • We can transform a given non-zero matrix to a simplified form called a Row-echelon form, using the row elementary operations. • We may have rows all of whose entries are zero. Such rows are called zero rows. For example, consider the following matrix.
  4. 4. Here R1 and R2 are non-zero rows. R3 is a zero row. A non-zero matrix A is said to be in a row-echelon form if: (i) All zero rows of A occur below every non-zero row of A. (ii) The first non-zero element in any row i of A occurs in the jth column of A, and then all other elements in the jth column of A below the first non-zero element of row i are zeros. (iii) The first on-zero entry in the ith row of A lies to the left of the first non-zero entry in ( i + 1)th row of A.
  5. 5. Now we apply elementary transformations. R2 → R2 – 2R1 R3 → R3 – 3R1 We get
  6. 6. R3 → R3 – 2R2 The above matrix is in row echelon form. Number of non-zero rows = 2 Hence the rank of matrix A = 2
  7. 7. Rank By Normal Form • Any matrix ‘A’ can be reduce to the following forms . A = [IR] or [Ir 0] • While ifinite number of elementory transformation these for are called normal form • Rank of the matrix by normal form is given by the order of identity matrix
  8. 8. Rank of a Matrix Using Normal Form Example:Find the rank of the matrix by reducing it to normal form Solution:
  9. 9. References • https://www.rgpvonline.com • https://byjus.com

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