Mathematics: (1 point) Supppose A is an invertible n×n matrix and v⃗ is an eigenvector of A with associated eigenvalue 8. Convince yourself that v⃗ is an eigenvector of the following matrices, and find the associated eigenvalues. The matrix A4 has an eigenvalue . The matrix A−1 has an eigenvalue . The matrix A+4In has an eigenvalue . The matrix 5A has an eigenvalue .

$2.03Step-by-step explanation:2.25-(2.25*.1)=2.025...

(1 point) Supppose A is an invertible n×n matrix

kelkel4731

24.01.2022

By definition, if $\vec v$ is an eigenvector of $\mathbf A$ (with associated eigenvalue 8), then

$\mathbf A\vec v=8\vec v$

Notice that

$\mathbf A^4\vec v=\mathbf A^3\cdot\mathbf A\vec v=8\mathbf A^3\vec v$

Rinse and repeat to find

$\mathbf A^4\vec v=8^4\vec v$

so that $\mathbf A^4$ has a corresponding eigenvalue of 4096.

Let $\lambda$ be the eigenvalue of $\mathbf A^{-1}$ corresponding to $\vec v$ . Then

$\mathbf A^{-1}\vec v=\lambda\vec v\implies\mathbf A\cdot\mathbf A^{-1}\vec v=\mathbf A\cdot\lambda \vec v\implies\vec v=8\lambda\vec v$

For this to hold, we require $8\lambda=1$ , or $\lambda=\frac18$ . So $\mathbf A^{-1}$ has a corresonding eigenvalue of 1/8.

Expanding gives

$(\mathbf A+4\mathbf I_n)\vec v=\mathbf A\vec v+4\mathbf I_n\vec v=8\vec v+4\vec v=12\vec v$

so that $\mathbf A+4\mathbf I_n$ has an associated eigenvalue of 12.

You know the drill:

$5\mathbf A\vec v=5\cdot8\vec v=40\vec v$

so the eigenvalue of $5\mathbf A$ is 40.

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