Question

If 2, 4, 6 are the eigen values of a 3 X 3 matrix A. Then the eigen values of AT are……

A. 1, 2, 3

B. 1, 4, 9

C. 1, 1/2, 1/3

D. 2, 4, 6

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Answer 1

Answer:

Just as 2 by 2 matrices can represent transformations of the plane, 3 by 3 matrices can represent transformations of 3D space. The picture is more complicated, but as in the 2 by 2 case, our best insights come from finding the matrix's eigenvectors: that is, those vectors whose direction the transformation leaves unchanged.

If non-zero e is an eigenvector of the 3 by 3 matrix A, then

Ae=e

for some scalar . This scalar is called an eigenvalue of A.

This may be rewritten

Ae=Ie

and in turn as

A−Ie=0

As in the 2 by 2 case, the matrix A−I must be singular. Once again, then, we ask: which are the values of for which A−I is singular? That is, the values that satisfy the characteristic equation

detA−I=0?

Consider the example

−2 −2 4 −4 1 2 2 2 5

The characteristic equation is

det −2− −2 4 −4 1− 2 2 2 5− =0

Expanding the determinant,

−2−[1−5−−22]+4[−25−−42]+2[−22−41−]=0

Expanding the brackets and simplifying:

−3+42+27−90=0

or, equivalently

3−42−27+90=0

By trial and error, we find that

33−432−273+90=0

and it follows from the Factor Theorem that −3 is a factor. Indeed,

3−42−27+90=−32−−30

and