Simplify the Matrix

Problem

[[1/2,1/2],[1/2,1/2]]*[[4,1],[4,1]]*[[1,−1],[1,1]]

Solution

1. Multiply the first two matrices.

[[1/2,1/2],[1/2,1/2]]*[[4,1],[4,1]]=[[(1/2)*(4)+(1/2)*(4),(1/2)*(1)+(1/2)*(1)],[(1/2)*(4)+(1/2)*(4),(1/2)*(1)+(1/2)*(1)]]

2. Simplify the resulting entries.

[[2+2,1/2+1/2],[2+2,1/2+1/2]]=[[4,1],[4,1]]

3. Multiply the result by the third matrix.

[[4,1],[4,1]]*[[1,−1],[1,1]]=[[(4)*(1)+(1)*(1),(4)*(−1)+(1)*(1)],[(4)*(1)+(1)*(1),(4)*(−1)+(1)*(1)]]

4. Simplify the final entries.

[[4+1,−4+1],[4+1,−4+1]]=[[5,−3],[5,−3]]

Final Answer

[[1/2,1/2],[1/2,1/2]]*[[4,1],[4,1]]*[[1,−1],[1,1]]=[[5,−3],[5,−3]]

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