Find the Variables

Problem

[[−3,−2,−1],[6,5,1],[−6,−2,2]]*[[a,b,c],[d,e,ƒ],[g,h,i]]=[[a,b,c],[d,e,ƒ],[g,h,i]]*[[5,0,0],[0,2,0],[0,0,3]]

Solution

1. Identify the matrix equation as an eigenvalue problem. The equation A*V=V*D where D is a diagonal matrix, implies that the columns of V are the eigenvectors of A corresponding to the diagonal entries of D

2. Set up the system for the first column [[a],[d],[g]] using the eigenvalue λ=5

(A−5*I)*(v_1)=0

[[−8,−2,−1],[6,0,1],[−6,−2,−3]]*[[a],[d],[g]]=[[0],[0],[0]]

3. Solve for a,d,g using row reduction. From the second row, g=−6*a Substituting into the first row: −8*a−2*d−(−6*a)=0⇒−2*a−2*d=0⇒d=−a Choosing a=1

[[a],[d],[g]]=[[1],[−1],[−6]]

4. Set up the system for the second column [[b],[e],[h]] using the eigenvalue λ=2

(A−2*I)*(v_2)=0

[[−5,−2,−1],[6,3,1],[−6,−2,0]]*[[b],[e],[h]]=[[0],[0],[0]]

5. Solve for b,e,h From the third row, −6*b−2*e=0⇒e=−3*b Substituting into the second row: 6*b+3*(−3*b)+h=0⇒−3*b+h=0⇒h=3*b Choosing b=1

[[b],[e],[h]]=[[1],[−3],[3]]

6. Set up the system for the third column [[c],[ƒ],[i]] using the eigenvalue λ=3

(A−3*I)*(v_3)=0

[[−6,−2,−1],[6,2,1],[−6,−2,−1]]*[[c],[ƒ],[i]]=[[0],[0],[0]]

7. Solve for c,ƒ,i All rows are multiples of 6*c+2*ƒ+i=0 We can choose c=1 and ƒ=−3 then 6*(1)+2*(−3)+i=0⇒i=0 Choosing c=1,ƒ=−3

[[c],[ƒ],[i]]=[[1],[−3],[0]]

Final Answer

[[a,b,c],[d,e,ƒ],[g,h,i]]=[[1,1,1],[−1,−3,−3],[−6,3,0]]

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