Simplify the Matrix

Problem

[[0,1,0],[1,−1,1],[−1,1,1]]*[[0,0,0],[0,18,0],[0,0,−18]]*[[1,1/2,−1/2],[1,0,0],[0,1/2,1/2]]

Solution

1. Multiply the first matrix by the second (diagonal) matrix. Multiplying a matrix A by a diagonal matrix D on the right scales the columns of A by the diagonal entries of D

[[0,1,0],[1,−1,1],[−1,1,1]]*[[0,0,0],[0,18,0],[0,0,−18]]=[[0,18,0],[0,−18,−18],[0,18,−18]]

2. Multiply the resulting matrix by the third matrix. Use the standard row-by-column multiplication rule.

[[0,18,0],[0,−18,−18],[0,18,−18]]*[[1,1/2,−1/2],[1,0,0],[0,1/2,1/2]]

3. Calculate the entries for the first row.

(R_1)⋅(C_1)=(0)*(1)+(18)*(1)+(0)*(0)=18

(R_1)⋅(C_2)=(0)*(1/2)+(18)*(0)+(0)*(1/2)=0

(R_1)⋅(C_3)=(0)*(−1/2)+(18)*(0)+(0)*(1/2)=0

4. Calculate the entries for the second row.

(R_2)⋅(C_1)=(0)*(1)+(−18)*(1)+(−18)*(0)=−18

(R_2)⋅(C_2)=(0)*(1/2)+(−18)*(0)+(−18)*(1/2)=−9

(R_2)⋅(C_3)=(0)*(−1/2)+(−18)*(0)+(−18)*(1/2)=−9

5. Calculate the entries for the third row.

(R_3)⋅(C_1)=(0)*(1)+(18)*(1)+(−18)*(0)=18

(R_3)⋅(C_2)=(0)*(1/2)+(18)*(0)+(−18)*(1/2)=−9

(R_3)⋅(C_3)=(0)*(−1/2)+(18)*(0)+(−18)*(1/2)=−9

Final Answer

[[0,1,0],[1,−1,1],[−1,1,1]]*[[0,0,0],[0,18,0],[0,0,−18]]*[[1,1/2,−1/2],[1,0,0],[0,1/2,1/2]]=[[18,0,0],[−18,−9,−9],[18,−9,−9]]

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