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If A=[[-1,2,3],[5,7,9],[-2,1,1]] and B=[...

If `A=[[-1,2,3],[5,7,9],[-2,1,1]]` and `B=[[-4,1,-5],[1,2,0],[1,3,1]]`, then verify that (i) (A+B)'=A'+B' (ii) (A-B)'=A'-B'

Text Solution

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`if A=[{:(-1,sqrt(3),2),(5,7,9),(-2,1,1):}]=[{:(-1,5,-2),(2,7,1),(3,9,1):}]`
` B=[{:(-4,1,-5),(1,2,0),(1,3,1):}]=[{:(-4,1,1),(1,2,3),(-5,0,1):}]`
`(i) A+B=[{:(-1,2,3),(5,7,9),(-2,1,1):}]+[{:(-4,1,-5),(1,2,0),(1,3,1):}]`
`=[{:(-5,3,-2),(6,9,9),(-1,4,2):}]`
`=[{:(-5 implies (A+B)'=[{:(-5,3,-2),(6,9,9),(-1,4,2):}]=[{:(-5,6,-1),(3,9,4),(-2,9,2):}]`
`and A' +B'=[{:(-1,5,-2),(2,7,1),(3,9,1):}]+[{:(-4,1,1),(1,2,3),(-5,0,1):}]`
`=[{:(-5,6,-1),(3,9,4),(-2,9,2):}]`
`therefore (A+B)'=A'+B'` hence proved
`(ii) A-B=[{:(-1,2,3),(5,7,9),(-2,1,1):}]-[{:(-4,1,-5),(1,2,0),(1,3,1):}]`
`=[{:(3,1,8),(4,5,9),(-3,-2,0):}]`
`implies (A-B)'=[{:(3,1,8),(4,5,9),(-3,-2,0):}]=[{:(3,4,-3),(1,5,-2),(8,9,0):}]`
` and A'-B' =[{:(-1,5,-2),(2,7,1),(3,9,1):}]-[{:(-4,1,1),(1,2,3),(-5,0,1):}]`
`=[{:(3,4,-3),(1,5,-2),(8,9,0):}]`
`therefore (A-B)'=A'B'` hence proved .
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