Multiply the Matrices

Problem

[[5,7,−3,−4],[7,6,−4,−5],[6,4,−3,−2],[8,5,−6,−1]]*[[7,8,6,9],[5,7,4,5],[3,4,5,6],[2,1,1,2]]

Solution

1. Identify the dimensions of the matrices. Both are 4×4 matrices, so the resulting product will also be a 4×4 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)=(5)*(7)+(7)*(5)+(−3)*(3)+(−4)*(2)=35+35−9−8=53

(R_1)*(C_2)=(5)*(8)+(7)*(7)+(−3)*(4)+(−4)*(1)=40+49−12−4=73

(R_1)*(C_3)=(5)*(6)+(7)*(4)+(−3)*(5)+(−4)*(1)=30+28−15−4=39

(R_1)*(C_4)=(5)*(9)+(7)*(5)+(−3)*(6)+(−4)*(2)=45+35−18−8=54

3. Calculate the elements of the second row.

(R_2)*(C_1)=(7)*(7)+(6)*(5)+(−4)*(3)+(−5)*(2)=49+30−12−10=57

(R_2)*(C_2)=(7)*(8)+(6)*(7)+(−4)*(4)+(−5)*(1)=56+42−16−5=77

(R_2)*(C_3)=(7)*(6)+(6)*(4)+(−4)*(5)+(−5)*(1)=42+24−20−5=41

(R_2)*(C_4)=(7)*(9)+(6)*(5)+(−4)*(6)+(−5)*(2)=63+30−24−10=59

4. Calculate the elements of the third row.

(R_3)*(C_1)=(6)*(7)+(4)*(5)+(−3)*(3)+(−2)*(2)=42+20−9−4=49

(R_3)*(C_2)=(6)*(8)+(4)*(7)+(−3)*(4)+(−2)*(1)=48+28−12−2=62

(R_3)*(C_3)=(6)*(6)+(4)*(4)+(−3)*(5)+(−2)*(1)=36+16−15−2=35

(R_3)*(C_4)=(6)*(9)+(4)*(5)+(−3)*(6)+(−2)*(2)=54+20−18−4=52

5. Calculate the elements of the fourth row.

(R_4)*(C_1)=(8)*(7)+(5)*(5)+(−6)*(3)+(−1)*(2)=56+25−18−2=61

(R_4)*(C_2)=(8)*(8)+(5)*(7)+(−6)*(4)+(−1)*(1)=64+35−24−1=74

(R_4)*(C_3)=(8)*(6)+(5)*(4)+(−6)*(5)+(−1)*(1)=48+20−30−1=37

(R_4)*(C_4)=(8)*(9)+(5)*(5)+(−6)*(6)+(−1)*(2)=72+25−36−2=59

Final Answer

[[5,7,−3,−4],[7,6,−4,−5],[6,4,−3,−2],[8,5,−6,−1]]*[[7,8,6,9],[5,7,4,5],[3,4,5,6],[2,1,1,2]]=[[53,73,39,54],[57,77,41,59],[49,62,35,52],[61,74,37,59]]

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