Simplify the Matrix

Problem

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

2. Simplify the arithmetic within the resulting matrix.

[[−2+0.5,−0.5+2],[2+0.5,0.5+2]]=[[−1.5,1.5],[2.5,2.5]]

3. Multiply this intermediate result by the third matrix.

[[−1.5,1.5],[2.5,2.5]]*[[−1,1],[1,1]]=[[(−1.5)*(−1)+(1.5)*(1),(−1.5)*(1)+(1.5)*(1)],[(2.5)*(−1)+(2.5)*(1),(2.5)*(1)+(2.5)*(1)]]

4. Calculate the final values for each entry in the matrix.

[[1.5+1.5,−1.5+1.5],[−2.5+2.5,2.5+2.5]]=[[3,0],[0,5]]

Final Answer

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

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