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Question
For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC):
`A=[[4 2 3],[1 1 2],[3 0 1]]`=`B=[[1 -1 1],[0 1 2],[2 -1 1]]` and `C= [[1 2 -1],[3 0 1],[0 0 1]]`
Solution
(AB)C=A(BC)
`⇒ ([[4 2 3],[1 1 2],[3 0 1]][[1 -1 1],[0 1 2],[2 -1 1]])` `[[1 2 -1],[3 0 1],[0 0 1]]=[[4 2 3],[1 1 2],[3 0 1]]` `([[1 -1 1],[0 1 2],[2 -1 1]] [[1 2 -1],[3 0 1],[0 0 1]])`
`⇒([[4+0+6 -4+2-3 4+4+3],[1+0+4 -1+1-2 1+2+2],[3+0+2 -3+0-1 3+0+1]])` `[[1 2 -1],[3 0 1],[0 0 1]]=[[4 2 3],[1 1 2],[3 0 1]]` `([[1-3+0 2-0+0 -1-1+1],[0+3+0 0+0+0 0+1+2],[2-3+0 4-0+0 -2-1+1]])`
⇒`[[10 -5 11],[5 -2 5],[5 -4 4]] [[1 2 -1],[3 0 1],[0 0 1]]=[[4 2 3],[1 1 2],[3 0 1]] [[-2 2 -1],[3 0 3],[-1 4 -2]]`
⇒`[[10-15+0 20-0+0 -10-5+11],[5-6+0 10-0+0 -5-2+5],[5-12+0 10-0+0 -5-4+4 ]]=[[-8+6-3 8+0+12 -4+6-6],[-2+3-2 2+0+8 -1+3-4],[-6+0-1 6+0+4 -3+0-2]]`
⇒`[[-5 20 -4],[-1 10 -2],[-7 10 -5]]` = `[[-5 20 -4],[-1 10 -2],[-7 10 -5]]`
∴ LHS=RHS
Hence proved.
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