If `A=[{:(0,-1,2),(4,3,-4):}]` and `B=[{:(4,0),(1,3),(2,6):}]` then verify that (i) (A')'=A (ii) (AB)'=B'A'

Explanation

we have, `A=[{:(0,-1,2),(4,3,-4):}]` and `B=[{:(4,0),(1,3),(2,6):}]`
(i) We have to verify that A'=A
`therefore A'=[{:(0,-1,2),(4,3,-4):}]=A`
(ii) We have to verify that AB=B A
`therefore AB=[{:(3,9),(11,-15):}]`
`rArr (AB)'=[{:(3,9),(9,15):}]`
and `B'A'= [{:(4,1,2),(0,3,6):}][{:(0,4),(-1,3),(2, -4):}]=[{:(3,11),(9,-15):}]`
=(AB)'
(iii) We have to verify that, (kA)=(kA)
Now, `(kA)=[{:(0,-k,2k),(4k,3k,-4k):}]`
and `(kA) ' = [{:(0,4k),(-k,3k),(2k,4k):}]`
=(kA)