Simplify the Matrix

Problem

[[3/7,1/7],[−3/7,6/7]]*[[1,−2],[−3,6]]*[[2,−1/3],[1,1]]

Solution

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

[[3/7,1/7],[−3/7,6/7]]*[[1,−2],[−3,6]]=[[(3/7)*(1)+(1/7)*(−3),(3/7)*(−2)+(1/7)*(6)],[(−3/7)*(1)+(6/7)*(−3),(−3/7)*(−2)+(6/7)*(6)]]

2. Simplify the entries of the resulting matrix.

[[3/7−3/7,−6/7+6/7],[−3/7−18/7,6/7+36/7]]=[[0,0],[−21/7,42/7]]=[[0,0],[−3,6]]

3. Multiply this intermediate result by the third matrix.

[[0,0],[−3,6]]*[[2,−1/3],[1,1]]=[[(0)*(2)+(0)*(1),(0)*(−1/3)+(0)*(1)],[(−3)*(2)+(6)*(1),(−3)*(−1/3)+(6)*(1)]]

4. Simplify the final entries to obtain the resulting matrix.

[[0+0,0+0],[−6+6,1+6]]=[[0,0],[0,7]]

Final Answer

[[3/7,1/7],[−3/7,6/7]]*[[1,−2],[−3,6]]*[[2,−1/3],[1,1]]=[[0,0],[0,7]]

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