Simplify the Matrix

Problem

[[0,−5,1],[1,4,−1],[0,1,0]]*[[−1,7,−1],[0,1,0],[0,15,2]]*[[1,1,1],[0,0,1],[0,1,5]]

Solution

1. Multiply the first two matrices together by calculating the dot product of each row of the first matrix with each column of the second matrix.

[[0,−5,1],[1,4,−1],[0,1,0]]*[[−1,7,−1],[0,1,0],[0,15,2]]=[[0*(−1)+(−5)*(0)+1*(0),0*(7)+(−5)*(1)+1*(15),0*(−1)+(−5)*(0)+1*(2)],[1*(−1)+4*(0)+(−1)*(0),1*(7)+4*(1)+(−1)*(15),1*(−1)+4*(0)+(−1)*(2)],[0*(−1)+1*(0)+0*(0),0*(7)+1*(1)+0*(15),0*(−1)+1*(0)+0*(2)]]

2. Simplify the resulting intermediate matrix.

[[0,10,2],[−1,−4,−3],[0,1,0]]

3. Multiply this intermediate matrix by the third matrix.

[[0,10,2],[−1,−4,−3],[0,1,0]]*[[1,1,1],[0,0,1],[0,1,5]]=[[0*(1)+10*(0)+2*(0),0*(1)+10*(0)+2*(1),0*(1)+10*(1)+2*(5)],[−1*(1)+(−4)*(0)+(−3)*(0),−1*(1)+(−4)*(0)+(−3)*(1),−1*(1)+(−4)*(1)+(−3)*(5)],[0*(1)+1*(0)+0*(0),0*(1)+1*(0)+0*(1),0*(1)+1*(1)+0*(5)]]

4. Calculate the final values for each entry.

[[0,2,20],[−1,−4,−20],[0,0,1]]

Final Answer

[[0,−5,1],[1,4,−1],[0,1,0]]*[[−1,7,−1],[0,1,0],[0,15,2]]*[[1,1,1],[0,0,1],[0,1,5]]=[[0,2,20],[−1,−4,−20],[0,0,1]]

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