Simplify the Matrix

Problem

[[2,−31,1],[4,7,−1],[1,2,−2]]*[[−15,0],[21,10],[13,1]]

Solution

1. Identify the dimensions of the matrices to ensure multiplication is possible. The first matrix is 3×3 and the second is 3×2 so the resulting matrix will be 3×2

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

(R_1)*(C_1)=(2)*(−15)+(−31)*(21)+(1)*(13)=−30−651+13=−668

(R_1)*(C_2)=(2)*(0)+(−31)*(10)+(1)*(1)=0−310+1=−309

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

(R_2)*(C_1)=(4)*(−15)+(7)*(21)+(−1)*(13)=−60+147−13=74

(R_2)*(C_2)=(4)*(0)+(7)*(10)+(−1)*(1)=0+70−1=69

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

(R_3)*(C_1)=(1)*(−15)+(2)*(21)+(−2)*(13)=−15+42−26=1

(R_3)*(C_2)=(1)*(0)+(2)*(10)+(−2)*(1)=0+20−2=18

5. Assemble the calculated values into the final 3×2 matrix.

Final Answer

[[2,−31,1],[4,7,−1],[1,2,−2]]*[[−15,0],[21,10],[13,1]]=[[−668,−309],[74,69],[1,18]]

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