Simplify the Matrix

Problem

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

Solution

1. Multiply the first two matrices together by calculating the dot product of rows and columns.

[[1/4,−1/2],[−1/4,−1/2]]*[[3,4],[1,3]]=[[(1/4)*(3)+(−1/2)*(1),(1/4)*(4)+(−1/2)*(3)],[(−1/4)*(3)+(−1/2)*(1),(−1/4)*(4)+(−1/2)*(3)]]

2. Simplify the arithmetic for each element in the resulting matrix.

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

3. Multiply this intermediate result by the third matrix.

[[1/4,−1/2],[−5/4,−5/2]]*[[2,−2],[1,1]]=[[(1/4)*(2)+(−1/2)*(1),(1/4)*(−2)+(−1/2)*(1)],[(−5/4)*(2)+(−5/2)*(1),(−5/4)*(−2)+(−5/2)*(1)]]

4. Simplify the final arithmetic to find the resulting matrix elements.

[[1/2−1/2,−1/2−1/2],[−5/2−5/2,5/2−5/2]]=[[0,−1],[−5,0]]

Final Answer

[[1/4,−1/2],[−1/4,−1/2]]*[[3,4],[1,3]]*[[2,−2],[1,1]]=[[0,−1],[−5,0]]

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