Simplify the Matrix

Problem

[[−1/2,1/2],[1/2,1/2]]*[[0,1],[1,0]]*[[−1,1],[1,1]]

Solution

1. Multiply the first two matrices.

[[−1/2,1/2],[1/2,1/2]]*[[0,1],[1,0]]=[[(−1/2)*(0)+(1/2)*(1),(−1/2)*(1)+(1/2)*(0)],[(1/2)*(0)+(1/2)*(1),(1/2)*(1)+(1/2)*(0)]]

[[0+1/2,−1/2+0],[0+1/2,1/2+0]]=[[1/2,−1/2],[1/2,1/2]]

2. Multiply the resulting matrix by the third matrix.

[[1/2,−1/2],[1/2,1/2]]*[[−1,1],[1,1]]=[[(1/2)*(−1)+(−1/2)*(1),(1/2)*(1)+(−1/2)*(1)],[(1/2)*(−1)+(1/2)*(1),(1/2)*(1)+(1/2)*(1)]]

3. Simplify the arithmetic for each element.

[[−1/2−1/2,1/2−1/2],[−1/2+1/2,1/2+1/2]]=[[−1,0],[0,1]]

Final Answer

[[−1/2,1/2],[1/2,1/2]]*[[0,1],[1,0]]*[[−1,1],[1,1]]=[[−1,0],[0,1]]

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