Essay About Rank R And Total Number

Canonical Matrix
Canonical matrix:             A non-zero matrix ‘A’ of rank r is row equivalent to a unique matrix C, called a canonical matrix of A, which is obtained from ‘A’ according to some definite rule. [Length of a matrix = total number of leading 1] ➢ Example: Find the canonical matrix that is row equivalent of the following matrix, A =[pic 2] We have,                             A =[pic 3] Performing R21 (-2), R31 (-3), R41 (-2), we get-[pic 4]         Performing R[pic 5]        , we get-[pic 6] Performing R12 (-2), R32 (-2), we get-[pic 7]         Performing R        [pic 8], we get-[pic 9]         Performing R13 (-5), R23 (1), we get-[pic 10]        = C Rank of A, ρ (A) = (maximum number of rows in A) – (number of zero rows in C) = 4 – 1 = 3,    & length = total number of leading ‘1’ = 3. Find the canonical matrix that is row equivalent of the following matrix– A = [pic 11] We have,  A = [pic 12] Performing R12, we get-[pic 13] Performing R21 (-2), R31 (-3), R41 (-4), we get-[pic 14]         Performing R        [pic 15], we get -[pic 16] Performing R12 (-2), R32 (-2), R42 (-5), we get-[pic 17]         Performing R        [pic 18], we get-[pic 19]         Performing R13 (-5), R23 (1), R43 (6), we get -[pic 20]        = C Rank of A, ρ (A) = (maximum number of rows in A) – (number of zero rows in C) = 4 – 1 = 3     & length = total number of leading ‘1’ = 3. Example: Find the canonical matrix that is row equivalent of the following matrix- A = [pic 21] We have, A = [pic 22] Performing R21 (-4), R31 (-6), we get-[pic 23]         Performing R        [pic 24], we get-[pic 25] Performing R12 (-2), R13 (5), we get -[pic 26]         Performing R        [pic 27], we get-[pic 28]