C1_W4_Quiz.md

1. Select the characteristic polynomial for the given matrix.

$$\begin{bmatrix} 2 & 1 \cr -3 & 6 \end{bmatrix}$$

• $\lambda^2 - 8 \lambda + 15$
• $\lambda^3 - 8 \lambda + 15$
• $\lambda^2 - 8 \lambda - 1$
• $\lambda^2 + 8 \lambda + 15$

2. Select the eigenvectors for the previous matrix in Q1, as given below:

$$\begin{bmatrix} 2 & 1 \cr -3 & 6 \end{bmatrix}$$

• $\pmatrix{1 \cr 0}, \pmatrix{0 \cr 1}$
• $\pmatrix{1 \cr 3}, \pmatrix{1 \cr 1}$
• $\pmatrix{1 \cr 3}, \pmatrix{1 \cr 3}$
• $\pmatrix{1 \cr 1}, \pmatrix{1 \cr 1}$

3. Which of the following is an eigenvalue for the given identity matrix.

$$ID = \begin{bmatrix} 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \end{bmatrix}$$

• $\lambda = 2$
• $\lambda = -1$
• $\lambda = 1$

4. Find the eigenvalues of matrix $A \cdot B$ where:

$$A = \begin{bmatrix} 1 & 2 \cr 0 & 4 \end{bmatrix}$$

$$B = \begin{bmatrix} 1 & 0 \cr 0 & 1 \end{bmatrix}$$

Hint: What type of matrix is B? Does it change the output when multiplied with A? If not, focus only on one of the matrices to find the eigenvalues.

• Eigenvalues cannot be determined.
• $\lambda_1 = 4, \lambda_2 = 2$
• $\lambda_1 = 3, \lambda_2 = 1$
• $\lambda_1 = 4, \lambda_2 = 1$

5. Select the eigenvectors, using the eigenvalues you found for the above matrix $A \cdot B$ in Q4.

• $\vec{v_1} = \left( 2, 0 \right); \vec{v_2} = \left( 1, 0 \right)$
• $\vec{v_1} = \left( 1, 3 \right); \vec{v_2} = \left( 1, 0 \right)$
• $\vec{v_1} = \left( 2, 3 \right); \vec{v_2} = \left( 1, 0 \right)$
• $\vec{v_1} = \left( 2, 3 \right); \vec{v_2} = \left( 2, 3 \right)$

6. Which of the vectors span the matrix $W$?

$$W = \begin{bmatrix} 2 & 3 & 0 \cr 1 & 2 & 5 \cr 3 & -2 & -1 \end{bmatrix}$$

• (A)

$$V_1 = \begin{bmatrix} 2 \cr 3 \cr 0 \end{bmatrix} V_2 = \begin{bmatrix} 1 \cr 2 \cr 5 \end{bmatrix} V_3 = \begin{bmatrix} 3 \cr -2 \cr -1 \end{bmatrix}$$

• (B)

$$V_1 = \begin{bmatrix} 2 \cr 1 \cr 3 \end{bmatrix} V_2 = \begin{bmatrix} 3 \cr 2 \cr -2 \end{bmatrix} V_3 = \begin{bmatrix} 0 \cr 5 \cr -1 \end{bmatrix}$$

7. Given matrix $P$ select the answer with the correct eigenbasis.

$$P = \begin{bmatrix} 2 & 0 & 0 \cr 1 & 2 & 1 \cr -1 & 0 & 1 \end{bmatrix}$$

Hint: First compute the eigenvalues, eigenvectors and contrust the eigenbasis matrix with the spanning eigenvectors.

• (A)

$$Eigenbasis = \begin{bmatrix} 0 & 0 & -1 \cr -1 & 1 & 0 \cr 1 & 0 & 1 \end{bmatrix}$$

• (B)

$$Eigenbasis = \begin{bmatrix} 0 & -1 & 1 \cr 0 & 1 & 0 \cr -1 & 0 & 1 \end{bmatrix}$$

• (C)

$$Eigenbasis = \begin{bmatrix} 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 1 \end{bmatrix}$$

8. Select the characteristic polynomial for the given matrix.

$$\begin{bmatrix} 3 & 1 & -2 \cr 4 & 0 & 1 \cr 2 & 1 & -1 \end{bmatrix}$$

• $\lambda^3 + 2 \lambda^2 + 4 \lambda - 5$
• $- \lambda^3 + 2 \lambda^2 + 9$
• $- \lambda^2 + 2 \lambda^3 + 4 \lambda - 5$
• $- \lambda^3 + 2 \lambda^2 + 4 \lambda - 5$

9. You are given a non-singular matrix $A$ with real entries and eigenvalue $i$.

Which of the following statements is correct?

• $1/i$ is an eigenvalue of $A^{-1}$.
• $i$ is an eigenvalue of $A^{-1} + A$.
• $i$ is an eigenvalue of $A^{-1} \cdot A \cdot I$.