Simplify the Matrix

Problem

[[−2,3,5],[2,−1,6],[4,−7,−3]]*[[7,−1,4],[−3,5,−2],[5,9,2]]

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

(c_11)=(−2)*(7)+(3)*(−3)+(5)*(5)=−14−9+25=2

(c_12)=(−2)*(−1)+(3)*(5)+(5)*(9)=2+15+45=62

(c_13)=(−2)*(4)+(3)*(−2)+(5)*(2)=−8−6+10=−4

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

(c_21)=(2)*(7)+(−1)*(−3)+(6)*(5)=14+3+30=47

(c_22)=(2)*(−1)+(−1)*(5)+(6)*(9)=−2−5+54=47

(c_23)=(2)*(4)+(−1)*(−2)+(6)*(2)=8+2+12=22

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

(c_31)=(4)*(7)+(−7)*(−3)+(−3)*(5)=28+21−15=34

(c_32)=(4)*(−1)+(−7)*(5)+(−3)*(9)=−4−35−27=−66

(c_33)=(4)*(4)+(−7)*(−2)+(−3)*(2)=16+14−6=24

Final Answer

[[−2,3,5],[2,−1,6],[4,−7,−3]]*[[7,−1,4],[−3,5,−2],[5,9,2]]=[[2,62,−4],[47,47,22],[34,−66,24]]

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