Q: A matrix $$A is $$said to be strictly diagonally dominant by rows if, for every row $$i$$, which of the following holds?

$$|a_{ii}| \u003e \sum_{j \neq i} |a_{ij}|$$

Question 1

Which of the following statements best describes the purpose of partial pivoting in Gaussian elimination?

Accepted Answer

To avoid division by zero and reduce the amplification of rounding errors by selecting the largest available pivot in each column.

Question 2

Given the system $$\begin{bmatrix} 1 & 1 & 1 \ 2 & 3 & 1 \ 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 6 \ 14 \ 1 \end{bmatrix}$$, what is the elimination multiplier $$m_{21}$$ used to eliminate the $$x$$ term from the second equation?

Accepted Answer

$$m_{21} = 2$$

Question 3

Suppose you have a matrix $$A$$ with $$\|A\|_\infty = 10$$ and $$\|A$$^{-1}$$\|_\infty = 50. $$What is the infinity-norm condition number $$\kappa_\infty(A)$$?

Accepted Answer

$$\kappa_\infty(A) = 500$$

Question 4

Which of the following is NOT a legal elementary row operation in Gaussian elimination?

Accepted Answer

Multiplying two rows together.

Question 5

A system $$Ax = b is $$solved using LU factorization with partial pivoting, resulting in $$PA = LU. $$What must be done to the right-hand side $$b$$ before solving the triangular systems?

Accepted Answer

Permute $$b$$ using the same permutation matrix $$P$$, so solve $$LUx = Pb.$$

Question 6

Consider the system $$\begin{bmatrix} 1 & 1 \ 1 & 1.0001 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 2 \ 2.0001 \end{bmatrix}. If $$the approximate solution is $$\hat{x} = [0, 2]^T$$, what is the infinity-norm residual?

Accepted Answer

$$0.0001$$

Question 7

Which iterative method updates each variable using only the values from the previous iteration (i.e., 'old' values)?

Accepted Answer

Jacobi method

Question 8

A matrix $$A is $$said to be strictly diagonally dominant by rows if, for every row $$i$$, which of the following holds?

Accepted Answer

$$|a_{ii}| \u003e \sum_{j 
eq i} |a_{ij}|$$

Question 9

In the Successive Over-Relaxation (SOR) method, what is the effect of choosing a relaxation parameter $$\omega \u003e 1$$?

Accepted Answer

It over-relaxes the update, moving beyond the Gauss–Seidel candidate.

Question 10

Which of the following best explains why a small residual does NOT always guarantee a small forward error in the solution of $$Ax = b$$?

Accepted Answer

Because the system may be ill-conditioned, so small changes in $$b$$ can cause large changes in $$x.$$

Question 11

Given $$A = \begin{bmatrix} 4 & 4 & 0 \ 2 & 2 & 1 \ 3 & 7 & 2 \end{bmatrix}$$, what is the first elimination multiplier $$m_{21}$$ for Gaussian elimination (no pivoting)?

Accepted Answer

$$m_{21} = 1/2$$

Question 12

Which of the following is the correct formula for the infinity norm of a vector $$v = [v_1$, v_2$, ..., v_n]^T$$?

Accepted Answer

$$\|v\|_\infty = \max_{1 \leq i \leq n} |v_i|$$

Question 13

If $$A is a$$ $$3 	imes 3$$ matrix and $$P is a $$permutation matrix representing a row swap, which of the following is true about $$P$$?

Accepted Answer

$$P is $$invertible and $$P^{-1} = P^T$.

Question 14

Which stopping criterion is most appropriate for an iterative solver for $$Ax = b$$?

Accepted Answer

Stop when $$\|b - Ax$$^{(k)}$$\|_\infty / \|b\|_\infty is $$less than a chosen tolerance.

Question 15

Which method is guaranteed to converge for any starting guess if the coefficient matrix is strictly diagonally dominant by rows?

Accepted Answer

Jacobi and Gauss–Seidel methods

Question 16

In the context of iterative methods, what is the main difference between Jacobi and Gauss–Seidel?

Accepted Answer

Jacobi uses only old values from the previous iteration, while Gauss–Seidel uses new values as soon as they are available.

Question 17

Suppose $$A is $$ill-conditioned. Which of the following is a reliable practice when solving $$Ax = b$$ numerically?

Accepted Answer

Compute the residual and interpret it using the condition number $$\kappa(A).$$

Question 18

Which of the following best describes the role of the lower triangular matrix $$L in $$the LU factorization $$A = LU$$?

Accepted Answer

$$L$$ stores the elimination multipliers used to create zeros below the diagonal in $$A.$$

Linear Systems: Direct and Iterative Methods in College Algebra

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