1.15 Eigenvalues and eigenvectors Quiz

1. Which equation defines an eigenvector $\mathbf{v}$ of $M$ with eigenvalue $\lambda$?

• A. $M\mathbf{v}=\mathbf{v}+\lambda$
• B. $\mathbf{v}M=\lambda\mathbf{0}$
• C. $M\mathbf{v}=\lambda\mathbf{v}$ with $\mathbf{v}\neq\mathbf{0}$
• D. $\det\mathbf{v}=\lambda$

2. Find the eigenvalues of $\begin{pmatrix}2&1\\ 1&2\end{pmatrix}$.

• A. $2$ and $3$
• B. $-1$ and $-3$
• C. $4$ and $3$
• D. $1$ and $3$

3. In $M=PDP^{-1}$, what are the columns of $P$?

• A. Eigenvectors of $M$, in the same order as the eigenvalues in $D$
• B. The eigenvalues of $M$
• C. The rows of $M$
• D. Any two independent vectors

4. $M$ has eigenvalues $2$ and $3$. What are the eigenvalues of $M^4$?

• A. $8$ and $12$
• B. $16$ and $81$
• C. $6$ and $4$
• D. $24$ only

5. A population model has $\mathbf{x}_n=c_1(1.1)^n\mathbf{v}_1+c_2(0.5)^n\mathbf{v}_2$ with $c_1\neq0$. What happens in the long term?

• A. Both populations die out
• B. The populations stay constant
• C. The populations grow in the ratio of $\mathbf{v}_1$
• D. The populations oscillate