Simplify the Matrix

Problem

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

Solution

1. Identify the dimensions of the matrices. Both are 3×3 matrices, so their product will also be a 3×3 matrix.

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

(R_1,1)=(0)*(1)+(9)*(8)+(3)*(1)=0+72+3=75

(R_1,2)=(0)*(4)+(9)*(9)+(3)*(−2)=0+81−6=75

(R_1,3)=(0)*(4)+(9)*(3)+(3)*(6)=0+27+18=45

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

(R_2,1)=(5)*(1)+(6)*(8)+(1)*(1)=5+48+1=54

(R_2,2)=(5)*(4)+(6)*(9)+(1)*(−2)=20+54−2=72

(R_2,3)=(5)*(4)+(6)*(3)+(1)*(6)=20+18+6=44

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

(R_3,1)=(3)*(1)+(8)*(8)+(4)*(1)=3+64+4=71

(R_3,2)=(3)*(4)+(8)*(9)+(4)*(−2)=12+72−8=76

(R_3,3)=(3)*(4)+(8)*(3)+(4)*(6)=12+24+24=60

5. Assemble the calculated values into the final product matrix.

Final Answer

[[0,9,3],[5,6,1],[3,8,4]]*[[1,4,4],[8,9,3],[1,−2,6]]=[[75,75,45],[54,72,44],[71,76,60]]

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