Differential Equations Quiz
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SYSTEMS OF DIFFERENTIAL EQUATIONS
Question 1
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Consider the system
x
′
=
A
x
\mathbf{x}' = A\mathbf{x}
x
′
=
A
x
where
A
=
(
5
−
2
8
−
3
)
A = \begin{pmatrix} 5 & -2 \\ 8 & -3 \end{pmatrix}
A
=
(
5
8
−
2
−
3
)
. If
λ
1
=
1
\lambda_1 = 1
λ
1
=
1
and
λ
2
=
1
\lambda_2 = 1
λ
2
=
1
are the eigenvalues, which statement about the corresponding eigenvectors is correct?
A
There is only one linearly independent eigenvector, so the matrix is defective and requires a generalized eigenvector
B
There are two linearly independent eigenvectors
v
1
=
(
1
2
)
\mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}
v
1
=
(
1
2
)
and
v
2
=
(
2
1
)
\mathbf{v}_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}
v
2
=
(
2
1
)
C
The eigenspace has dimension 2, making the matrix diagonalizable with complete eigenspace
D
The repeated eigenvalue guarantees two distinct eigenvectors in different eigenspaces
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