Simplify the Matrix

Problem

[[−(3√(,5))/25,(14√(,5))/25],[−(2√(,205))/25,√(,205)/25]]*[[0,2],[2,3]]*[[1/√(,5),−14/√(,205)],[2/√(,5),−3/√(,205)]]

Solution

1. Multiply the first two matrices together by calculating the dot product of each row from the first matrix with each column of the second matrix.

(M_12)=[[(−(3√(,5))/25)*(0)+((14√(,5))/25)*(2),(−(3√(,5))/25)*(2)+((14√(,5))/25)*(3)],[(−(2√(,205))/25)*(0)+(√(,205)/25)*(2),(−(2√(,205))/25)*(2)+(√(,205)/25)*(3)]]

2. Simplify the entries of the resulting matrix (M_12)

(M_12)=[[(28√(,5))/25,(36√(,5))/25],[(2√(,205))/25,−√(,205)/25]]

3. Multiply the resulting matrix (M_12) by the third matrix.

(M_ƒ*i*n*a*l)=[[((28√(,5))/25)*(1/√(,5))+((36√(,5))/25)*(2/√(,5)),((28√(,5))/25)*(−14/√(,205))+((36√(,5))/25)*(−3/√(,205))],[((2√(,205))/25)*(1/√(,5))+(−√(,205)/25)*(2/√(,5)),((2√(,205))/25)*(−14/√(,205))+(−√(,205)/25)*(−3/√(,205))]]

4. Simplify the arithmetic for each entry. For the top-left: 28/25+72/25=100/25=4 For the bottom-left: (2√(,205))/(25√(,5))−(2√(,205))/(25√(,5))=0 For the bottom-right: −28/25+3/25=−25/25=−1

(M_ƒ*i*n*a*l)=[[4,(−392√(,5)−108√(,5))/(25√(,205))],[0,−1]]

5. Evaluate the top-right entry. Note that √(,205)=√(,5⋅41) The numerator is −500√(,5)

Top-right=(−500√(,5))/(25√(,205))=(−20√(,5))/(√(,5)√(,41))=−20/√(,41)

Final Answer

[[−(3√(,5))/25,(14√(,5))/25],[−(2√(,205))/25,√(,205)/25]]*[[0,2],[2,3]]*[[1/√(,5),−14/√(,205)],[2/√(,5),−3/√(,205)]]=[[4,−20/√(,41)],[0,−1]]

Want more problems? Check here!