Multiply the Matrices

Problem

[[4,3,−1],[3,0,1],[−1,−2,1]]*[[12,−19/2,37/2],[−8,8,−12],[−20,17,−31]]

Solution

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

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.

(R_1)*(C_1)=(4)*(12)+(3)*(−8)+(−1)*(−20)=48−24+20=44

(R_1)*(C_2)=(4)*(−19/2)+(3)*(8)+(−1)*(17)=−38+24−17=−31

(R_1)*(C_3)=(4)*(37/2)+(3)*(−12)+(−1)*(−31)=74−36+31=69

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.

(R_2)*(C_1)=(3)*(12)+(0)*(−8)+(1)*(−20)=36+0−20=16

(R_2)*(C_2)=(3)*(−19/2)+(0)*(8)+(1)*(17)=−57/2+17=−57/2+34/2=−23/2

(R_2)*(C_3)=(3)*(37/2)+(0)*(−12)+(1)*(−31)=111/2−31=111/2−62/2=49/2

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.

(R_3)*(C_1)=(−1)*(12)+(−2)*(−8)+(1)*(−20)=−12+16−20=−16

(R_3)*(C_2)=(−1)*(−19/2)+(−2)*(8)+(1)*(17)=19/2−16+17=19/2+1=19/2+2/2=21/2

(R_3)*(C_3)=(−1)*(37/2)+(−2)*(−12)+(1)*(−31)=−37/2+24−31=−37/2−7=−37/2−14/2=−51/2

Final Answer

[[4,3,−1],[3,0,1],[−1,−2,1]]*[[12,−19/2,37/2],[−8,8,−12],[−20,17,−31]]=[[44,−31,69],[16,−23/2,49/2],[−16,21/2,−51/2]]

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