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Chapter 7: Q46E (page 383)

TRUE OR FALSE

If vector v鈫抜s an eigenvector of both A and B, then v鈫抜s an eigenvector of AB.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define eigenvector

A vector is said to be an eigenvector if, after being subjected to a linear transformation, its direction does not change.

02

Explanation for v→  is an eigenvector of AB

Consider v鈫� is an eigenvector of both A and B, then

Av鈫� = 位v鈫�

Bv鈫� = 碌v鈫�

Now,

ABv鈫� = A(碌v鈫�)

= 碌Av鈫�

= 位碌v鈫�

So, v鈫� is an eigenvector of AB.

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