Eigenvalues and eigenvectors Quiz

1. What are the eigenvalues of $\begin{pmatrix} 1 & 2 \\ 3 & 0 \end{pmatrix}$?

• A. $-3$ and $2$
• B. $1$ and $0$
• C. $3$ and $-2$
• D. $-1$ and $6$

2. $\mathbf{A}$ has eigenvalue $3$ with eigenvector $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$. What is $\mathbf{A}\begin{pmatrix} 2 \\ 4 \end{pmatrix}$?

• A. $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$
• B. $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$
• C. $\begin{pmatrix} 6 \\ 12 \end{pmatrix}$
• D. $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$

3. What is the normalised form of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$?

• A. $\begin{pmatrix} \tfrac{3}{7} \\ \tfrac{4}{7} \end{pmatrix}$
• B. $\begin{pmatrix} \tfrac{3}{5} \\ \tfrac{4}{5} \end{pmatrix}$
• C. $\begin{pmatrix} \tfrac{3}{25} \\ \tfrac{4}{25} \end{pmatrix}$
• D. $\begin{pmatrix} \tfrac{1}{5} \\ \tfrac{1}{5} \end{pmatrix}$

4. A $3\times3$ matrix has eigenvalues $1$, $2$ and $5$. What is its determinant?

• A. $8$
• B. $10$
• C. $7$
• D. $0$

5. A $2\times2$ matrix has an eigenvalue $0$. Which statement must be true?

• A. $\mathbf{A}=\mathbf{0}$
• B. Its trace is $0$
• C. $\mathbf{A}$ is the identity
• D. $\det\mathbf{A}=0$