Eigenvalues and eigenvectors Quiz

1. Find the eigenvalues of $\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$.

• A. $1$ and $6$
• B. $3$ and $4$
• C. $2$ and $5$
• D. $-2$ and $-5$

2. Which vector is an eigenvector of $\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ with eigenvalue $3$?

• A. $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$
• B. $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$
• C. $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$
• D. $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$

3. A matrix has eigenvalue $-2$ with eigenvector $\mathbf{v}$. What does the transformation do to points on the line through $O$ in the direction of $\mathbf{v}$?

• A. Reflects them in the line
• B. Rotates them through $180^\circ$ about $O$, keeping the distance from $O$
• C. Leaves them unchanged
• D. Stretches them by scale factor 2 and moves them to the opposite side of $O$

4. What is the characteristic equation of a $3\times3$ matrix?

• A. A quadratic in $\lambda$
• B. A cubic in $\lambda$
• C. A linear equation in $\lambda$
• D. An equation with no $\lambda$ term

5. For a $3\times3$ matrix with eigenvalue $1$ and eigenvector $\mathbf{v}$, the points on the line through $O$ in the direction of $\mathbf{v}$ are:

• A. reflected in the line
• B. stretched by scale factor 1 from a different centre
• C. unchanged (invariant)
• D. mapped to the origin