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Problem

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

Solution

1. Identify the operation as matrix multiplication of two 3×3 matrices.

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

(c_11)=(1)*(−2)+(−1)*(0)+(−2)*(0)=−2

(c_12)=(1)*(1)+(−1)*(1)+(−2)*(0)=0

(c_13)=(1)*(−1)+(−1)*(2)+(−2)*(1)=−5

3. Calculate the second row of the resulting matrix by taking the dot product of the second row of the left matrix with each column of the right matrix.

(c_21)=(0)*(−2)+(2)*(0)+(3)*(0)=0

(c_22)=(0)*(1)+(2)*(1)+(3)*(0)=2

(c_23)=(0)*(−1)+(2)*(2)+(3)*(1)=7

4. Calculate the third row of the resulting matrix by taking the dot product of the third row of the left matrix with each column of the right matrix.

(c_31)=(0)*(−2)+(0)*(0)+(1)*(0)=0

(c_32)=(0)*(1)+(0)*(1)+(1)*(0)=0

(c_33)=(0)*(−1)+(0)*(2)+(1)*(1)=1

5. Assemble the calculated values into the final product matrix.

Final Answer

[[1,−1,−2],[0,2,3],[0,0,1]]*[[−2,1,−1],[0,1,2],[0,0,1]]=[[−2,0,−5],[0,2,7],[0,0,1]]

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