Question

Suppose A is an invertible n×n matrix and v is an eigenvector of A with associated eigenvalue 4. Convince yourself that v is an eigenvector of the following matrices, and find the associated eigenvalues:
A4, eigenvalue = ?
A−1, eigenvalue = ?
A+4In, eigenvalue = ?
2A, eigenvalue = ?

Answer 1

The eigenvector of matrix A4 is 4096.

The eigenvalue of the matrix A-1 is 1/8.

(A+4I) has an eigenvalue of 12.

40 is the eigenvalue of 5Av.

The collection of scalar values known as the eigenvalues of a matrix are connected to the set of linear equations that are most likely contained within the matrix equations.

The eigenvectors are also known as the characteristic roots.

If the matrix A's eigen vector v is linked to an eigen value.

Av then equals lamda v.

The fact that v is an eigen vector of A with the value eight is assumed.

hence, Av = 8v.

We must determine the eigenvalue of A4.

A4*v equals A3(8v) = 84*v equals 4096v.

As a result, the eigenvalue of A4 is 4096.

Suppose A has an eigenvalue of 8. The eigenvalue of A-1 is thus 1/8.

It is calculated that the eigenvalue of (A+4I) is (A +4In)v = Av + 4v = 8v + 4v =12v.

(A+4I) has an eigenvalue of 12.

The value of 5Av's eigenvalue is 5Av = 5*8v = 40v.

40 is the eigenvalue of 5Av.

Learn more about the eigen vectors here:

brainly.com/question/15423383

#SPJ4