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]