Simplify the Matrix

Problem

[[−1,−1,−3],[−1,1,0],[1,0,1]]*[[3,1,3],[1,3,3],[−1,−1,−1]]*[[1,1,3],[1,2,3],[−1,−1,−2]]

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.

A=[[−1,−1,−3],[−1,1,0],[1,0,1]]

B=[[3,1,3],[1,3,3],[−1,−1,−1]]

A*B=[[(−1)*(3)+(−1)*(1)+(−3)*(−1),(−1)*(1)+(−1)*(3)+(−3)*(−1),(−1)*(3)+(−1)*(3)+(−3)*(−1)],[(−1)*(3)+(1)*(1)+(0)*(−1),(−1)*(1)+(1)*(3)+(0)*(−1),(−1)*(3)+(1)*(3)+(0)*(−1)],[(1)*(3)+(0)*(1)+(1)*(−1),(1)*(1)+(0)*(3)+(1)*(−1),(1)*(3)+(0)*(3)+(1)*(−1)]]

2. Simplify the resulting entries of the product A*B

A*B=[[−3−1+3,−1−3+3,−3−3+3],[−3+1+0,−1+3+0,−3+3+0],[3+0−1,1+0−1,3+0−1]]

A*B=[[−1,−1,−3],[−2,2,0],[2,0,2]]

3. Multiply the result A*B by the third matrix C

C=[[1,1,3],[1,2,3],[−1,−1,−2]]

(A*B)*C=[[(−1)*(1)+(−1)*(1)+(−3)*(−1),(−1)*(1)+(−1)*(2)+(−3)*(−1),(−1)*(3)+(−1)*(3)+(−3)*(−2)],[(−2)*(1)+(2)*(1)+(0)*(−1),(−2)*(1)+(2)*(2)+(0)*(−1),(−2)*(3)+(2)*(3)+(0)*(−2)],[(2)*(1)+(0)*(1)+(2)*(−1),(2)*(1)+(0)*(2)+(2)*(−1),(2)*(3)+(0)*(3)+(2)*(−2)]]

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

(A*B)*C=[[−1−1+3,−1−2+3,−3−3+6],[−2+2+0,−2+4+0,−6+6+0],[2+0−2,2+0−2,6+0−4]]

(A*B)*C=[[1,0,0],[0,2,0],[0,0,2]]

Final Answer

[[−1,−1,−3],[−1,1,0],[1,0,1]]*[[3,1,3],[1,3,3],[−1,−1,−1]]*[[1,1,3],[1,2,3],[−1,−1,−2]]=[[1,0,0],[0,2,0],[0,0,2]]

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