Suppose that v is an eigenvector of matrix a with eigenvalue λa, and it is also an eigenvector of matrix b with eigenvalue λb. (a) show that v is an eigenvector of a + b and find its associated eigenvalue. (b) show that v is an eigenvector of ab and find its associated eigenvalue.

(a) Yes,

(b) Yes,

Step-by-step explanation:

First, lets understand what are eigenvectors and eigenvalues?

Note: I am using the notation to denote Lambda(A) sign.

is an eigenvector of matrix A with eigenvalue

is also eigenvector of matrix B with eigenvalue

So we can write this in equation form as

So what does this equation say?

When you multiply any vector by A they do change their direction. any vector  that is in the same direction as of , then this  is called the eigenvector of . is times the original . The number is the eigenvalue of A.

this number is very important and tells us what is happening when we multiply . Is it shrinking or expanding or reversed or something else?

It tells us everything we need to know!

Bonus:

By the way you can find out the eigenvalue of by using the following equation:

where I is identity matrix of the size of same as A.

Now lets come to the solution!

(a) Show that is an eigenvector of and find its associated eigenvalue.

The eigenvalues of and are and , then

so,  

which means that is also an eigenvector of and the associated eigenvalues are

(b) Show that is an eigenvector of and find its associated eigenvalue.

The eigenvalues of and are and , then

so,  

which means that is also an eigenvector of and the associated eigenvalues are

false

step-by-step explanation: