Simplify the Matrix

Problem

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

Solution

1. Identify the dimensions of the matrices. Both are 3×3 matrices, so the resulting matrix will also be 3×3

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

(c_11)=(−1)*(0)+(0)*(1)+(−1)*(1)=−1

(c_12)=(−1)*(0)+(0)*(2)+(−1)*(0)=0

(c_13)=(−1)*(−2)+(0)*(1)+(−1)*(3)=2−3=−1

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

(c_21)=(0)*(0)+(1)*(1)+(−1)*(1)=1−1=0

(c_22)=(0)*(0)+(1)*(2)+(−1)*(0)=2

(c_23)=(0)*(−2)+(1)*(1)+(−1)*(3)=1−3=−2

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

(c_31)=(1)*(0)+(0)*(1)+(2)*(1)=2

(c_32)=(1)*(0)+(0)*(2)+(2)*(0)=0

(c_33)=(1)*(−2)+(0)*(1)+(2)*(3)=−2+6=4

5. Assemble the resulting matrix using the calculated values.

Final Answer

[[−1,0,−1],[0,1,−1],[1,0,2]]*[[0,0,−2],[1,2,1],[1,0,3]]=[[−1,0,−1],[0,2,−2],[2,0,4]]

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