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Question
For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC):
`A =-[[1 2 0],[-1 0 1]]`,`B=[[1 0],[-1 2],[0 3]]` and C= `[[1],[-1]]`
Solution
(AB)C=A(BC)
`⇒([[1 2 0],[-1 0 1]][[1 0],[-1 2],[0 3]])` `[[1],[-1]]=[[1 2 0],[-1 0 1]]` `([[1 0],[-1 2],[0 3]] [[1],[-1]])`
`⇒([[1-2+0 0+4+0],[-1-0+0 0+0+3]])` `[[1],[-1]]=[[1 2 0],[-1 0 1]]``([[1 -0],[-1 -2],[0 -3]])`
`⇒ [[-1 4],[-1 3]] [[1],[-1]]=[[1 2 0],[-1 0 1]][[[1],[-3],[-3]]]`
`⇒[[-1 -4],[-1 -3]]=[[1-6-0],[-1-0-3]]`
`⇒[[-5],[-4]]=[[-5],[-4]]`
∴ LHS=RHS
Hence proved.
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