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Chapter 7: Q30E (page 372)

(a). If 2i is an eigenvalue of a real 2 脳 2 matrix A, findA2.

(b). Give an example of a real 2 脳 2 matrix A such that all the entries of A are nonzero and 2i is an eigenvalue of A. ComputeA2and check that your answer agrees with part (a).

Short Answer

Expert verified

(a). The solution for A2= -4I.

(b). The solution for A2= -4I . Thus, it agrees with part (a) answer.

Step by step solution

01

Define eigenvalue:

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue, often denoted by位.

02

Find A2:

Given the eigenvalue of A is 2i, then the other eigenvalue is -2i.

So the equation is ,

fA(位)=(位鈭�2i)(位+2i)

=位2+4

From this we get,

A2+4I=0

Calculating for A2,

role="math" localid="1664255493746" A2=-4I

03

Find and compare if it agrees with part (a):

So we get,

fA(位)=(1鈭�位)(鈭�1鈭�位)+5

=位2+4

So in part (a),

位=2iis an eigenvalue of A.

A2=鈭�400鈭�4

A2=-4I, so this agrees with part (a).

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