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Problem

[[2,−3,6],[0,1,−2],[5,7,4]]*[[1,7,0,5],[12,3,2,−4],[3,−4,8,9]]

Solution

1. Identify the dimensions of the matrices to ensure multiplication is possible. The first matrix is 3×3 and the second is 3×4 so the resulting matrix will be 3×4

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)=(2)*(1)+(−3)*(12)+(6)*(3)=2−36+18=−16

(R_1)*(C_2)=(2)*(7)+(−3)*(3)+(6)*(−4)=14−9−24=−19

(R_1)*(C_3)=(2)*(0)+(−3)*(2)+(6)*(8)=0−6+48=42

(R_1)*(C_4)=(2)*(5)+(−3)*(−4)+(6)*(9)=10+12+54=76

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)=(0)*(1)+(1)*(12)+(−2)*(3)=0+12−6=6

(R_2)*(C_2)=(0)*(7)+(1)*(3)+(−2)*(−4)=0+3+8=11

(R_2)*(C_3)=(0)*(0)+(1)*(2)+(−2)*(8)=0+2−16=−14

(R_2)*(C_4)=(0)*(5)+(1)*(−4)+(−2)*(9)=0−4−18=−22

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)=(5)*(1)+(7)*(12)+(4)*(3)=5+84+12=101

(R_3)*(C_2)=(5)*(7)+(7)*(3)+(4)*(−4)=35+21−16=40

(R_3)*(C_3)=(5)*(0)+(7)*(2)+(4)*(8)=0+14+32=46

(R_3)*(C_4)=(5)*(5)+(7)*(−4)+(4)*(9)=25−28+36=33

Final Answer

[[2,−3,6],[0,1,−2],[5,7,4]]*[[1,7,0,5],[12,3,2,−4],[3,−4,8,9]]=[[−16,−19,42,76],[6,11,−14,−22],[101,40,46,33]]

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