Multiply the Matrices

Problem

[[3,6],[2,−3],[−2,4],[3,−1]]*[[4,10,15],[15,13,16]]

Solution

1. Identify the dimensions of the matrices to ensure multiplication is possible. The first matrix is 4×2 and the second is 2×3 Since the number of columns in the first equals the number of rows in the second, the resulting matrix will be 4×3

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.

(R_1)*(C_1)=(3)*(4)+(6)*(15)=12+90=102

(R_1)*(C_2)=(3)*(10)+(6)*(13)=30+78=108

(R_1)*(C_3)=(3)*(15)+(6)*(16)=45+96=141

3. Calculate the elements of the second row.

(R_2)*(C_1)=(2)*(4)+(−3)*(15)=8−45=−37

(R_2)*(C_2)=(2)*(10)+(−3)*(13)=20−39=−19

(R_2)*(C_3)=(2)*(15)+(−3)*(16)=30−48=−18

4. Calculate the elements of the third row.

(R_3)*(C_1)=(−2)*(4)+(4)*(15)=−8+60=52

(R_3)*(C_2)=(−2)*(10)+(4)*(13)=−20+52=32

(R_3)*(C_3)=(−2)*(15)+(4)*(16)=−30+64=34

5. Calculate the elements of the fourth row.

(R_4)*(C_1)=(3)*(4)+(−1)*(15)=12−15=−3

(R_4)*(C_2)=(3)*(10)+(−1)*(13)=30−13=17

(R_4)*(C_3)=(3)*(15)+(−1)*(16)=45−16=29

6. Assemble the calculated values into the final 4×3 matrix.

Final Answer

[[3,6],[2,−3],[−2,4],[3,−1]]*[[4,10,15],[15,13,16]]=[[102,108,141],[−37,−19,−18],[52,32,34],[−3,17,29]]

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