MTH603 Quiz 2 Fall 2024

Here are the MCQs and their answers from the quiz:

1. Differences methods find the ________ solution of the system.
   Answer: numerical

2. Two matrices with the same characteristic polynomial need not be similar.
   Answer: TRUE

3. A and its transpose matrix have _______ eigenvalues.
   Answer: same

4. Differences methods are iterative methods.
   Answer: FALSE

5. While using power method, the computed vector $u(1) = \begin{pmatrix} 7 \\ 12 \\ 14 \end{pmatrix}$ will be in normalized form as:
   Answer: $u(1) = \frac{1}{17} \begin{pmatrix} 7/17 \\ 12/17 \\ 14/17 \end{pmatrix}$

6. The Jacobi iteration ______, if A is strictly diagonally dominant.
   Answer: converges

7. Central difference method seems to be giving a better approximation, however it requires more computations.
   Answer: TRUE

8. The characteristic polynomial of a 3x3 identity matrix is __________:
   Answer: $(x - 1)^3$

9. While using Jacobi method for the matrix

   $$A = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$

   and 'theta $\theta = \frac{\pi}{4}$', the orthogonal matrix $S_1$ is:
   Answer:

   $$S_1 = \begin{pmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4} & 0 \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

10. Direct methods can be more rapid than iterative algorithms.
    Answer: TRUE

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