Multiply the Matrices

Problem

[[−1,−2/5,1],[−4,1,1],[1,0,0]]*[[2,2,8],[5,−1,−7],[0,0,2]]*[[−1,−2/5,1],[−4,1,1],[1,0,0]]^(−1)

Solution

1. Identify the matrices as P D and P^(−1) where the expression is in the form P*D*P^(−1)

2. Calculate the inverse of the first matrix P=[[−1,−2/5,1],[−4,1,1],[1,0,0]]

3. Find the determinant of P using the third row.

det(P)=1⋅((−2/5⋅1)−(1⋅1))=−7/5

4. Determine the adjugate matrix adj(P) and multiply by 1/det(P) to find P^(−1)

P^(−1)=[[0,0,1],[−1,1,3],[1,2/5,13/5]]

5. Multiply the first two matrices P and D

P*D=[[−1,−2/5,1],[−4,1,1],[1,0,0]]*[[2,2,8],[5,−1,−7],[0,0,2]]=[[−4,−8/5,−14],[−3,−9,−37],[2,2,8]]

6. Multiply the result (P*D) by P^(−1)

(P*D)*P^(−1)=[[−4,−8/5,−14],[−3,−9,−37],[2,2,8]]*[[0,0,1],[−1,1,3],[1,2/5,13/5]]=[[−62/5,−36/5,−226/5],[−28,−119/5,−616/5],[−2,26/5,144/5]]

7. Simplify the resulting matrix elements.

(P*D)*P^(−1)=[[−12.4,−7.2,−45.2],[−28,−23.8,−123.2],[−2,5.2,28.8]]

Final Answer

[[−1,−2/5,1],[−4,1,1],[1,0,0]]*[[2,2,8],[5,−1,−7],[0,0,2]]*[[−1,−2/5,1],[−4,1,1],[1,0,0]]^(−1)=[[−62/5,−36/5,−226/5],[−28,−119/5,−616/5],[−2,26/5,144/5]]

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