Simplify the Matrix

Problem

[[3,−1,4],[5,−3,4],[0,4,−2]]*[[5,1,5],[4,4,2],[3,−1,4]]

Solution

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

2. Calculate the elements of the first row. Multiply the first row of the left matrix by each column of the right matrix.

(R_1)*(C_1)=(3)*(5)+(−1)*(4)+(4)*(3)=15−4+12=23

(R_1)*(C_2)=(3)*(1)+(−1)*(4)+(4)*(−1)=3−4−4=−5

(R_1)*(C_3)=(3)*(5)+(−1)*(2)+(4)*(4)=15−2+16=29

3. Calculate the elements of the second row. Multiply the second row of the left matrix by each column of the right matrix.

(R_2)*(C_1)=(5)*(5)+(−3)*(4)+(4)*(3)=25−12+12=25

(R_2)*(C_2)=(5)*(1)+(−3)*(4)+(4)*(−1)=5−12−4=−11

(R_2)*(C_3)=(5)*(5)+(−3)*(2)+(4)*(4)=25−6+16=35

4. Calculate the elements of the third row. Multiply the third row of the left matrix by each column of the right matrix.

(R_3)*(C_1)=(0)*(5)+(4)*(4)+(−2)*(3)=0+16−6=10

(R_3)*(C_2)=(0)*(1)+(4)*(4)+(−2)*(−1)=0+16+2=18

(R_3)*(C_3)=(0)*(5)+(4)*(2)+(−2)*(4)=0+8−8=0

5. Assemble the resulting matrix using the calculated values.

Final Answer

[[3,−1,4],[5,−3,4],[0,4,−2]]*[[5,1,5],[4,4,2],[3,−1,4]]=[[23,−5,29],[25,−11,35],[10,18,0]]

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