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