Multiply the Matrices

Problem

[[2,4,3],[1,3,1],[5,2,2],[2,4,5],[1,1,1]]*[[2,1,5,2,1],[4,3,2,4,1],[3,1,2,5,1]]

Solution

1. Identify the dimensions of the matrices. The first matrix is 5×3 and the second matrix is 3×5 The resulting matrix will be 5×5

2. Calculate the entries for the first row of the product by multiplying the first row of the left matrix by each column of the right matrix.

(R_1)*(C_1)=(2)*(2)+(4)*(4)+(3)*(3)=4+16+9=29

(R_1)*(C_2)=(2)*(1)+(4)*(3)+(3)*(1)=2+12+3=17

(R_1)*(C_3)=(2)*(5)+(4)*(2)+(3)*(2)=10+8+6=24

(R_1)*(C_4)=(2)*(2)+(4)*(4)+(3)*(5)=4+16+15=35

(R_1)*(C_5)=(2)*(1)+(4)*(1)+(3)*(1)=2+4+3=9

3. Calculate the entries for the second row.

(R_2)*(C_1)=(1)*(2)+(3)*(4)+(1)*(3)=2+12+3=17

(R_2)*(C_2)=(1)*(1)+(3)*(3)+(1)*(1)=1+9+1=11

(R_2)*(C_3)=(1)*(5)+(3)*(2)+(1)*(2)=5+6+2=13

(R_2)*(C_4)=(1)*(2)+(3)*(4)+(1)*(5)=2+12+5=19

(R_2)*(C_5)=(1)*(1)+(3)*(1)+(1)*(1)=1+3+1=5

4. Calculate the entries for the third row.

(R_3)*(C_1)=(5)*(2)+(2)*(4)+(2)*(3)=10+8+6=24

(R_3)*(C_2)=(5)*(1)+(2)*(3)+(2)*(1)=5+6+2=13

(R_3)*(C_3)=(5)*(5)+(2)*(2)+(2)*(2)=25+4+4=33

(R_3)*(C_4)=(5)*(2)+(2)*(4)+(2)*(5)=10+8+10=28

(R_3)*(C_5)=(5)*(1)+(2)*(1)+(2)*(1)=5+2+2=9

5. Calculate the entries for the fourth row.

(R_4)*(C_1)=(2)*(2)+(4)*(4)+(5)*(3)=4+16+15=35

(R_4)*(C_2)=(2)*(1)+(4)*(3)+(5)*(1)=2+12+5=19

(R_4)*(C_3)=(2)*(5)+(4)*(2)+(5)*(2)=10+8+10=28

(R_4)*(C_4)=(2)*(2)+(4)*(4)+(5)*(5)=4+16+25=45

(R_4)*(C_5)=(2)*(1)+(4)*(1)+(5)*(1)=2+4+5=11

6. Calculate the entries for the fifth row.

(R_5)*(C_1)=(1)*(2)+(1)*(4)+(1)*(3)=2+4+3=9

(R_5)*(C_2)=(1)*(1)+(1)*(3)+(1)*(1)=1+3+1=5

(R_5)*(C_3)=(1)*(5)+(1)*(2)+(1)*(2)=5+2+2=9

(R_5)*(C_4)=(1)*(2)+(1)*(4)+(1)*(5)=2+4+5=11

(R_5)*(C_5)=(1)*(1)+(1)*(1)+(1)*(1)=1+1+1=3

Final Answer

[[2,4,3],[1,3,1],[5,2,2],[2,4,5],[1,1,1]]*[[2,1,5,2,1],[4,3,2,4,1],[3,1,2,5,1]]=[[29,17,24,35,9],[17,11,13,19,5],[24,13,33,28,9],[35,19,28,45,11],[9,5,9,11,3]]

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