Ordinary Differential Equations
MATH 308 - 523
A. Bonito
April 14
Spring 2015
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Quiz 5
• 5 minute individual quiz;
• Answer the questions in the space provided. If you run out of space, continue
onto the back of the page. Additional space is provided at the end;
• Show and explain all work;
• Underline the answer of each steps;
• The use of books, personal notes, calculator, cellphone, laptop, and communication with others is forbidden;
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Exercise 1
100%
Find the eigenvalues and one associated eigenvector (for each eigenvalue) of the matrix below and
determine whether they are linearly independent.
−3 3/4
−5 1
Ordinary Differential Equations
MATH 308 - 523
A. Bonito
April 14
Spring 2015
Quiz 5: solutions
Exercise 1
100%
The eigenvalues are given by
det(A − λI) = 0.
This is
det
−3 − λ
−5
3/4
1−λ
=0
or computing the determinant
λ2 + 2λ + 3/4 = 0
The solutions of the above equation are given by
λ1 = −
3
λ2 = − .
2
1
2
The eigenvectors ξ 1 = (ξ1 , ξ2 )t associated with λ1 satisfy
3
−5ξ1 + ξ2 = 0
2
or
ξ1 = α
1
3
− 10
10
−3
for any constant α. For instance
ξ1 =
The eigenvectors ξ 2 = (ξ1 , ξ2 )t associated with λ2 satisfy
3
−3ξ1 + ξ2 = 0
2
or
2
ξ =α
1
1
2
for any constant α. For instance,
2
ξ =
2
1
.
The two eigenvectors are independent because they correspond to different eigenvalues.