Simplify the Matrix

Problem

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

Solution

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

[[1/2,−1/2],[−1/2,3/2]]*[[1,3],[−1,5]]=[[(1/2)*(1)+(−1/2)*(−1),(1/2)*(3)+(−1/2)*(5)],[(−1/2)*(1)+(3/2)*(−1),(−1/2)*(3)+(3/2)*(5)]]

2. Simplify the arithmetic for each entry in the resulting 2×2 matrix.

[[1/2+1/2,3/2−5/2],[−1/2−3/2,−3/2+15/2]]=[[1,−1],[−2,6]]

3. Multiply this intermediate result by the third matrix.

[[1,−1],[−2,6]]*[[3,1],[1,1]]=[[(1)*(3)+(−1)*(1),(1)*(1)+(−1)*(1)],[(−2)*(3)+(6)*(1),(−2)*(1)+(6)*(1)]]

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

[[3−1,1−1],[−6+6,−2+6]]=[[2,0],[0,4]]

Final Answer

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

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