Numerical Linear Algebra Quizzes: Test Your Understanding

Question 1

Which of the following methods is most suitable for solving a system of linear equations with a diagonally dominant coefficient matrix?

Accepted Answer

Gauss-Seidel iteration

Answer

Jacobi iteration

Answer

LU decomposition

Question 2

Which method is specifically designed for solving linear systems with sparse coefficient matrices?

Accepted Answer

Conjugate gradient method

Answer

Gaussian elimination

Answer

LU decomposition

Answer

Power iteration

Question 3

What is the computational complexity of Gaussian elimination with partial pivoting for an n x n matrix?

Accepted Answer

O(n^3)

Answer

O(n^2)

Answer

O(n^4)

Answer

O(n^5)

Question 4

What is the primary purpose of employing a preconditioner in iterative methods for solving linear systems?

Accepted Answer

Improving the rate of convergence

Answer

Reducing the number of iterations required

Answer

Ensuring the stability of the method

Answer

Reducing the memory requirements

Question 5

Which iterative method is specifically designed for finding the smallest eigenvalue of a symmetric positive definite matrix?

Accepted Answer

Lanczos method

Answer

Power iteration

Answer

Inverse iteration

Answer

QR iteration

Question 6

In numerical linear algebra, for a symmetric positive definite matrix, which matrix decomposition technique is used to solve a system of linear equations?

Accepted Answer

Cholesky Decomposition

Answer

QR Decomposition

Answer

LU Decomposition

Answer

Eigenvalue Decomposition

Question 7

Which of the following is NOT a fundamental operation in numerical linear algebra?

Accepted Answer

Integration of matrix

Answer

Matrix multiplication

Answer

Vector subtraction

Answer

Matrix inversion

Question 8

What is the primary purpose of an eigenvalue decomposition?

Accepted Answer

To determine the eigenvectors and eigenvalues of a matrix

Answer

To solve a system of linear equations

Answer

To find the inverse of a matrix

Answer

To calculate the rank of a matrix

Question 9

What is the computational complexity of Gaussian elimination for solving an n x n linear system?

Accepted Answer

O(n^3)

Answer

O(n)

Answer

O(n^2)

Answer

O(2^n)

Question 10

What is the definition of the condition number of a matrix?

Accepted Answer

A measure of how sensitive a matrix is to minor changes

Answer

The number of eigenvalues of a matrix

Answer

The determinant of a matrix

Answer

The trace of a matrix

Question 11

Which of the following is NOT a type of matrix decomposition?

Accepted Answer

Eigenvector decomposition

Answer

LU decomposition

Answer

QR decomposition

Answer

Cholesky decomposition

Question 12

What is the primary purpose of a singular value decomposition (SVD)?

Accepted Answer

To reduce the dimensionality of data

Answer

To find the eigenvectors and eigenvalues of a matrix

Answer

To find the null space of a matrix

Answer

To find the rank of a matrix

Question 13

What is the fundamental difference between a sparse matrix and a dense matrix?

Accepted Answer

A sparse matrix has primarily zero entries, while a dense matrix has primarily non-zero entries

Answer

A sparse matrix is always symmetric, while a dense matrix is not

Answer

A sparse matrix is always invertible, while a dense matrix is not

Answer

A sparse matrix is always positive definite, while a dense matrix is not

Question 14

What is the definition of an eigenvalue of a matrix?

Accepted Answer

A scalar λ such that (A - λI)x = 0 has a non-trivial solution for some vector x.

Answer

A vector v such that Av = λv for some scalar λ.

Answer

The largest element of the matrix A.

Answer

The smallest element of the matrix A.

Question 15

What is the computational complexity of LU decomposition for an n x n matrix?

Accepted Answer

O(n^3)

Answer

O(n)

Answer

O(n^2)

Answer

O(n^4)

Question 16

Which application best exemplifies the use of QR decomposition?

Accepted Answer

Solving least squares problems

Answer

Computing eigenvalues and eigenvectors

Answer

Finding the rank of a matrix

Answer

Inverting a matrix

Question 17

What is the key difference between a sparse matrix and a dense matrix?

Accepted Answer

A sparse matrix has predominantly zero-valued elements, whereas a dense matrix has mostly non-zero elements.

Answer

A sparse matrix is square, while a dense matrix is rectangular.

Answer

A sparse matrix is symmetric, while a dense matrix is not.

Answer

A sparse matrix is invertible, while a dense matrix is not.

Question 18

What does the condition number of a matrix represent?

Accepted Answer

A measure of how sensitive the solution to a system of linear equations is to changes in the matrix

Answer

A measure of the size of the matrix

Answer

A measure of the determinant of the matrix

Answer

A measure of the rank of the matrix

Question 19

What is the greatest eigenvalue of the matrix A = [[2, 1], [-1, 2]]?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 20

The condition number of a matrix measures its:

Accepted Answer

Stability

Answer

Accuracy

Answer

Sparsity

Answer

Rank

Question 21

The QR algorithm is primarily used to find:

Accepted Answer

Eigenvalues of a matrix

Answer

Inverse of a matrix

Answer

Determinant of a matrix

Answer

Rank of a matrix

Question 22

Which of the following is a characteristic of a sparse matrix?

Accepted Answer

Contains mostly zero entries

Answer

Contains all non-zero entries

Answer

Has a large number of rows and columns

Answer

Has a high condition number

Question 23

Which of the following is NOT a type of matrix decomposition?

Accepted Answer

Gaussian Elimination

Answer

LU Decomposition

Answer

QR Decomposition

Answer

Singular Value Decomposition

Question 24

For a system of linear equations with a nonsingular coefficient matrix, which method guarantees an exact solution?

Accepted Answer

Gauss-Jordan elimination

Answer

Cramer's rule

Answer

Jacobi iteration

Answer

Power iteration

Question 25

For finding eigenvalues of a symmetric matrix, which decomposition method is specifically tailored?

Accepted Answer

Cholesky Decomposition

Answer

QR Decomposition

Answer

Power Iteration

Answer

LU Decomposition

Question 26

For solving a system of linear equations with a symmetric and positive definite coefficient matrix, which decomposition method is the most suitable?

Accepted Answer

Cholesky Decomposition

Answer

LU Decomposition

Answer

QR Decomposition

Answer

Power Iteration

Question 27

Which of the following is a primary benefit of using QR factorization for solving large linear systems?

Accepted Answer

It is more efficient than Gaussian elimination for large, sparse, or ill-conditioned matrices.

Answer

It provides higher accuracy than Gaussian elimination.

Answer

It can be used to compute eigenvalues and eigenvectors more efficiently.

Answer

It is more stable than Gaussian elimination for solving systems with small or zero pivots.

Question 28

What is an eigenvalue of a matrix A? Choose the correct definition.

Accepted Answer

It is a scalar that scales the corresponding eigenvector when multiplied by A

Answer

It is the determinant of A

Answer

It is the largest element of A

Answer

It is the number of non-zero rows in A

Question 29

Which matrix decomposition can be used to find the inverse of a non-singular matrix?

Accepted Answer

LU decomposition

Answer

Singular Value Decomposition

Answer

Cholesky decomposition

Answer

QR decomposition

Question 30

What does the condition number of a matrix represent?

Accepted Answer

A measure of how close a matrix is to being singular

Answer

The number of non-zero eigenvalues of a matrix

Answer

The determinant of a matrix

Answer

The trace of a matrix

Question 31

Which of the following is NOT an application of numerical linear algebra in image processing?

Accepted Answer

Statistical analysis

Answer

Image compression

Answer

Image enhancement

Answer

Feature detection

Question 32

Which matrix decomposition is used to solve a least squares problem?

Accepted Answer

Singular Value Decomposition (SVD)

Answer

Cholesky decomposition

Answer

LU decomposition

Answer

QR decomposition

Question 33

What does the condition number of a matrix measure?

Accepted Answer

The sensitivity of a matrix to perturbations

Answer

The accuracy of a matrix inversion

Answer

The rank of a matrix

Answer

The determinant of a matrix

Question 34

Which of the following describes the condition number of a matrix A?

Accepted Answer

The ratio of the largest singular value to the smallest singular value

Answer

The determinant of A

Answer

The trace of A

Question 35

Which representation is used for sparse matrices to store only non-zero elements?

Accepted Answer

Compressed Sparse Row (CSR) format

Answer

Band matrix format

Answer

Dense matrix format

Answer

Tridiagonal matrix format

Question 36

What is an advantage of using iterative methods for solving linear systems?

Accepted Answer

Can handle large, sparse systems efficiently

Answer

Are easier to implement than direct methods

Answer

Always converge to the exact solution

Answer

Are more accurate than direct methods

Question 37

Which numerical method is specifically designed to solve systems of linear equations with sparse matrices, where most of the elements are zero?

Accepted Answer

Krylov subspace methods

Answer

LU decomposition

Answer

QR decomposition

Answer

Gaussian elimination

Question 38

Define the eigenvalue of a matrix. Express the mathematical equation that relates an eigenvalue (λ) to a corresponding non-zero vector (x).

Accepted Answer

Ax = λx

Answer

A^T x = λx

Answer

x^T A = λx

Answer

Ax = 0

Question 39

The condition number of a matrix is a significant indicator of its:

Accepted Answer

Sensitivity to perturbations

Answer

Matrix determinant

Answer

Matrix size

Answer

Matrix rank

Question 40

Which technique is commonly employed to enhance the accuracy of numerical solutions, particularly in iterative methods?

Accepted Answer

Iterative refinement

Answer

Matrix scaling

Answer

Truncation error estimation

Answer

Round-off error analysis

Question 41

The QR decomposition expresses a matrix as a product of:

Accepted Answer

Orthogonal and triangular matrices

Answer

Symmetric and positive definite matrices

Answer

Triangular and diagonal matrices

Answer

Eigenvectors and eigenvalues

Question 42

The power method is an iterative technique specifically designed to approximate:

Accepted Answer

The dominant eigenvalue

Answer

The smallest eigenvalue

Answer

The trace of the matrix

Answer

The condition number

Question 43

Which type of matrix is guaranteed to be diagonalizable?

Accepted Answer

Symmetric matrix

Answer

Triangular matrix

Answer

Orthogonal matrix

Answer

Skew-symmetric matrix

Question 44

What is a primary purpose of the condition number of a matrix?

Accepted Answer

To estimate the sensitivity of a matrix to small changes in its elements

Answer

To determine the determinant of a matrix

Answer

To check if a matrix is invertible

Answer

To find the eigenvalues of a matrix

Question 45

Which fundamental difference distinguishes direct methods from iterative methods for solving linear systems?

Accepted Answer

Direct methods find exact solutions in a finite number of steps, while iterative methods approach solutions gradually through successive approximations

Answer

Direct methods are more efficient for large systems, while iterative methods are faster for sparse systems.

Question 46

What is the primary purpose of pivoting in Gauss-Jordan elimination?

Accepted Answer

To maintain numerical stability by preventing division by small numbers, which can lead to large errors

Answer

To transform a matrix into row echelon form

Answer

To find the determinant of a matrix

Question 47

Which algorithm is commonly used to compute the singular value decomposition (SVD) of a matrix?

Accepted Answer

Golub-Reinsch Algorithm

Answer

Jacobi Method

Answer

QR Algorithm

Answer

Power Method

Question 48

A symmetric matrix is characterized by which properties of its eigenvalues and eigenvectors?

Accepted Answer

Eigenvectors are orthogonal, and eigenvalues are real

Answer

Eigenvectors are orthogonal, and eigenvalues are complex

Answer

Eigenvectors are linearly independent, and eigenvalues are complex

Question 49

The condition number of a matrix A, denoted as cond(A), is a measure of:

Accepted Answer

How sensitive the solution of a linear system Ax = b is to small changes in A or b.

Answer

The determinant of matrix A.

Answer

The number of eigenvalues of matrix A.

Question 50

Given a matrix A, its eigenvalues λ and eigenvectors v, which of the following equations correctly represents the relationship between them?

Accepted Answer

Av = λv

Answer

A + v = λ

Answer

A - v = λ

Question 51

Which of the following matrix decompositions is commonly used for efficiently solving linear systems with a symmetric positive definite matrix?

Accepted Answer

Cholesky Decomposition

Answer

LU Decomposition

Answer

QR Decomposition

Answer

Singular Value Decomposition (SVD)

Question 52

What is the primary advantage of using QR factorization compared to Gaussian elimination for solving linear systems?

Accepted Answer

QR factorization is generally more numerically stable, especially for ill-conditioned matrices.

Answer

QR factorization is simpler to implement than Gaussian elimination.

Answer

QR factorization is consistently faster than Gaussian elimination.

Question 53

Which of the following methods is commonly used to find the eigenvalues of a large, sparse matrix, where computational efficiency is crucial?

Accepted Answer

Power Method

Answer

Cholesky Decomposition

Answer

LU Decomposition

Answer

Gaussian Elimination

Question 54

What is the primary purpose of using an iterative method like the Jacobi method or Gauss-Seidel method for solving linear systems?

Accepted Answer

To iteratively approximate the solution, especially when dealing with large systems where finding an exact solution might be computationally expensive.

Answer

To find the exact solution of the system in a single step.

Answer

To solve systems with non-linear equations.

Question 55

Which of the following statements accurately describes the singular value decomposition (SVD) of a matrix A?

Accepted Answer

SVD decomposes A into the product of three matrices: U, Σ, and V^T, where U and V are orthogonal matrices and Σ is a diagonal matrix containing singular values.

Answer

SVD is only applicable to square matrices.

Question 56

A matrix is considered ill-conditioned if its condition number is:

Accepted Answer

Large

Answer

Small

Answer

Zero

Answer

Negative

Question 57

Which of the following techniques is NOT typically used for solving eigenvalue problems?

Accepted Answer

LU Decomposition

Answer

QR Algorithm

Answer

Inverse Iteration

Answer

Power Method