MATH152 April 2016

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### Question A 24

A ${\displaystyle 3\times 3}$ matrix A with real entries has been typed into MATLAB. The result of the command = eig(A) is (after some slight formatting changes to make it fit better in the exam):

V = 0.8165 + 0.0000i   0.8165 + 0.0000i   0.5774 + 0.0000i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.5774 + 0.0000i
0.4082 - 0.4082i   0.4082 + 0.4082i   0.5774 + 0.0000i

D = 1.0000 + 2.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0000 + 0.0000i   1.0000 - 2.0000i   0.0000 + 0.0000i
0.0000 + 0.0000i   0.0000 + 0.0000i  -1.0000 + 0.0000i

Circle all true statements below:

(a)
A has no real eigenvalues.
(b)
All eigenvalues of A have negative real parts.
(c)
The eigenvectors of A are a basis for ${\displaystyle \mathbb {R} ^{3}}$.
(d)
Eigenvectors of A associated to distinct complex eigenvalues are linearly independent.
(e)
${\displaystyle [1,1,1]^{T}}$ is an eigenvector of A.
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