Simplify the Matrix

Problem

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

Solution

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

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

(R_1)*(C_1)=(3)*(1)+(−5)*(2)+(1)*(3)=3−10+3=−4

(R_1)*(C_2)=(3)*(0)+(−5)*(5)+(1)*(−5)=0−25−5=−30

(R_1)*(C_3)=(3)*(3)+(−5)*(1)+(1)*(3)=9−5+3=7

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

(R_2)*(C_1)=(0)*(1)+(−2)*(2)+(3)*(3)=0−4+9=5

(R_2)*(C_2)=(0)*(0)+(−2)*(5)+(3)*(−5)=0−10−15=−25

(R_2)*(C_3)=(0)*(3)+(−2)*(1)+(3)*(3)=0−2+9=7

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

(R_3)*(C_1)=(0)*(1)+(5)*(2)+(−1)*(3)=0+10−3=7

(R_3)*(C_2)=(0)*(0)+(5)*(5)+(−1)*(−5)=0+25+5=30

(R_3)*(C_3)=(0)*(3)+(5)*(1)+(−1)*(3)=0+5−3=2

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

Final Answer

[[3,−5,1],[0,−2,3],[0,5,−1]]*[[1,0,3],[2,5,1],[3,−5,3]]=[[−4,−30,7],[5,−25,7],[7,30,2]]

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