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Problem

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

Solution

1. Identify the dimensions of the matrices. The first matrix is 4×3 and the second matrix is 3×4 The resulting matrix will be 4×4

2. Calculate the entries of the first row by taking the dot product of the first row of the left matrix with each column of the right matrix.

(R_1)*(C_1)=(2)*(2)+(0)*(0)+(1)*(1)=5

(R_1)*(C_2)=(2)*(−1)+(0)*(1)+(1)*(0)=−2

(R_1)*(C_3)=(2)*(1)+(0)*(3)+(1)*(1)=3

(R_1)*(C_4)=(2)*(−1)+(0)*(1)+(1)*(1)=−1

3. Calculate the entries of the second row.

(R_2)*(C_1)=(−1)*(2)+(1)*(0)+(0)*(1)=−2

(R_2)*(C_2)=(−1)*(−1)+(1)*(1)+(0)*(0)=2

(R_2)*(C_3)=(−1)*(1)+(1)*(3)+(0)*(1)=2

(R_2)*(C_4)=(−1)*(−1)+(1)*(1)+(0)*(1)=2

4. Calculate the entries of the third row.

(R_3)*(C_1)=(1)*(2)+(3)*(0)+(1)*(1)=3

(R_3)*(C_2)=(1)*(−1)+(3)*(1)+(1)*(0)=2

(R_3)*(C_3)=(1)*(1)+(3)*(3)+(1)*(1)=11

(R_3)*(C_4)=(1)*(−1)+(3)*(1)+(1)*(1)=3

5. Calculate the entries of the fourth row.

(R_4)*(C_1)=(−1)*(2)+(1)*(0)+(1)*(1)=−1

(R_4)*(C_2)=(−1)*(−1)+(1)*(1)+(1)*(0)=2

(R_4)*(C_3)=(−1)*(1)+(1)*(3)+(1)*(1)=3

(R_4)*(C_4)=(−1)*(−1)+(1)*(1)+(1)*(1)=3

Final Answer

[[2,0,1],[−1,1,0],[1,3,1],[−1,1,1]]*[[2,−1,1,−1],[0,1,3,1],[1,0,1,1]]=[[5,−2,3,−1],[−2,2,2,2],[3,2,11,3],[−1,2,3,3]]

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