Numerical Linear Algebra Quizzes: Master Matrix Computations and Eigenvalues

Question 1

Which of the following matrices is non-invertible?

Accepted Answer

[[1, 2], [0, 0]]

Answer

[[1, 2], [3, 4]]

Answer

[[1, 1], [1, 1]]

Answer

[[0, 1], [1, 0]]

Question 2

What is the rank of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 3

What is matrix conditioning and why is it important in solving systems of linear equations?

Accepted Answer

Matrix conditioning measures how sensitive the solution of a linear system is to changes in its input data, indicating the stability and accuracy of numerical solutions.

Answer

Matrix conditioning determines the number of solutions to a system of linear equations.

Answer

Matrix conditioning is a measure of the largest eigenvalue of a matrix.

Question 4

What iterative method is commonly used to solve linear equations?

Accepted Answer

Gauss-Seidel method

Answer

Newton-Raphson method

Answer

Bisection method

Answer

Lagrange multiplier method

Question 5

What is the computational complexity of LU decomposition for a dense matrix?

Accepted Answer

O(n^3)

Answer

O(n^2)

Answer

O(n^4)

Answer

O(n^5)

Question 6

Which matrix norm measures the maximum singular value?

Accepted Answer

Spectral norm

Answer

Frobenius norm

Answer

Infinity norm

Answer

1-norm

Question 7

Which iterative method is suitable for solving linear equations with symmetric positive definite matrices?

Accepted Answer

Conjugate gradient method

Answer

Gauss-Seidel method

Answer

QR algorithm

Answer

Power iteration

Question 8

How do the null space and column space of a matrix differ?

Accepted Answer

The null space consists of vectors perpendicular to the column space, while the column space spans the vectors formed by the matrix's columns.

Answer

The column space consists of vectors perpendicular to the null space, while the null space spans the vectors formed by the matrix's rows.

Answer

The null space and column space are always orthogonal subspaces.

Answer

The null space and column space have the same dimension.

Question 9

Consider the statement: "The numerical rank of a matrix is the maximum number of linearly independent rows or columns it contains." Is this statement true or false?

Accepted Answer

True

Answer

False

Question 10

What is the purpose of a matrix condition number?

Accepted Answer

To measure the sensitivity of a solution to changes in the matrix.

Answer

To measure the size of the matrix.

Answer

To measure the determinant of the matrix.

Answer

To measure the eigenvalues of the matrix.

Question 11

Which numerical method is commonly used to solve large sparse linear systems?

Accepted Answer

Conjugate gradient method

Answer

LU decomposition

Answer

QR decomposition

Answer

Gaussian elimination

Question 12

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

Accepted Answer

Polynomial fitting

Answer

Matrix multiplication

Answer

Solving linear equations

Answer

Finding eigenvalues and eigenvectors

Question 13

What is the rank of the matrix [1 2; 3 4; 5 6]?

Accepted Answer

2

Answer

1

Answer

3

Question 14

What is a characteristic of a positive definite matrix?

Accepted Answer

All eigenvalues are positive

Answer

All entries on the diagonal are positive

Answer

The determinant is positive

Answer

The trace is positive

Question 15

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

Accepted Answer

Integration

Answer

Matrix computations

Answer

Solving linear equations

Answer

Analyzing eigenvalues and eigenvectors

Question 16

How does QR decomposition help solve linear least squares problems?

Accepted Answer

Transforms the problem into an orthogonal system to solve for the coefficients.

Answer

Finds the eigenvectors and eigenvalues of the coefficient matrix.

Answer

Calculates the null space of the coefficient matrix.

Question 17

What is the primary goal of applying Gaussian elimination to a matrix?

Accepted Answer

Transforming the matrix into row echelon form

Answer

Solving a system of linear equations

Answer

Inverting the matrix

Answer

Computing eigenvalues

Question 18

Which real-world application of Numerical Linear Algebra is used extensively in computer graphics?

Accepted Answer

Matrix transformations for 3D objects

Answer

Solving partial differential equations

Answer

Physical system simulation

Answer

Financial data analysis

Question 19

Which class of methods is specifically designed for solving large systems of linear equations?

Accepted Answer

Iterative methods

Answer

QR decomposition

Answer

Gaussian elimination

Answer

LU factorization

Question 20

Which decomposition expresses a matrix as the product of an orthogonal matrix and an upper triangular matrix?

Accepted Answer

QR decomposition

Answer

Cholesky decomposition

Answer

LU decomposition

Question 21

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

Accepted Answer

To decompose a matrix into a product of three matrices, revealing its underlying structure

Answer

To compute eigenvalues and eigenvectors

Answer

To solve systems of linear equations

Question 22

Which operation is excluded from the fundamental operations in numerical linear algebra?

Accepted Answer

Differentiation

Answer

Eigenvalue analysis

Answer

Solving linear equations

Answer

Matrix computations

Question 23

What is the primary objective of performing an LU decomposition?

Accepted Answer

Efficiently resolving systems of linear equations.

Answer

Calculating the rank of a matrix.

Answer

Determining a matrix's determinant.

Answer

Identifying the eigenvalues of a matrix.

Question 24

What significance does the condition number of a matrix hold?

Accepted Answer

Reveals the sensitivity of the solution to changes in the matrix.

Answer

Gives the determinant of the matrix.

Answer

Indicates the number of linearly independent rows in the matrix.

Question 25

Name an application of numerical linear algebra within the realm of data analysis.

Accepted Answer

Principal component analysis

Answer

Trigonometry

Answer

Calculus

Question 26

In the context of an n x n matrix, what is the computational complexity of Gaussian elimination?

Accepted Answer

O(n^3)

Answer

O(n)

Answer

O(n log n)

Answer

O(n^2)

Question 27

Identify a type of iterative method employed for solving linear equations.

Accepted Answer

Jacobi iteration

Answer

QR factorization

Answer

Gaussian elimination with pivoting

Answer

LU decomposition

Question 28

In the context of machine learning, what is the role of linear algebra?

Accepted Answer

Supplies the foundational mathematical framework for numerous machine learning algorithms.

Answer

No direct relation to machine learning.

Answer

Solely used during the data preprocessing stage of machine learning.

Question 29

Find the inverse of the matrix B = [[1,0],[0,1]]

Accepted Answer

[[1,0],[0,1]]

Answer

[[2,0],[0,2]]

Answer

[[0,1],[1,0]]

Answer

[[-1,0],[0,-1]]

Question 30

Which of the following is the adjoint of the matrix D = [[1,2],[3,4]]?

Accepted Answer

[[4,-2],[-3,1]]

Answer

[[1,2],[-3,4]]

Answer

[[4,2],[-3,1]]

Answer

[[1,-2],[3,-4]]

Question 31

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

Accepted Answer

Differentiation

Answer

Multiplication

Answer

Addition

Answer

Transpose

Question 32

What is the primary purpose of LU decomposition?

Accepted Answer

To solve systems of linear equations efficiently

Answer

To find eigenvalues of a matrix

Answer

To compute the determinant of a matrix

Answer

To invert a matrix

Question 33

What is the condition number of a matrix used to represent?

Accepted Answer

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

Answer

The number of eigenvalues of a matrix.

Question 34

Which of the following methods is an iterative method for solving linear systems of equations?

Accepted Answer

Gauss-Seidel method

Answer

LU decomposition

Answer

Cholesky factorization

Answer

Gaussian elimination

Question 35

What is the condition number of a matrix and what information does it provide about the matrix?

Accepted Answer

It is a measure of how sensitive the solution of a linear system is to changes in the input data. A high condition number indicates that the matrix is ill-conditioned.

Answer

It is the determinant of the matrix, indicating its singularity or non-singularity.

Question 36

Which of the following algorithms is commonly used to find eigenvalues and eigenvectors of a symmetric matrix?

Accepted Answer

QR algorithm

Answer

Jacobi iteration

Answer

Power iteration method

Question 37

Given a square matrix A and its inverse A⁻¹, which of the following statements is TRUE?

Accepted Answer

The product of A and A⁻¹ equals the identity matrix.

Answer

The determinant of A⁻¹ is equal to the determinant of A.

Answer

The inverse of a matrix always exists.

Question 38

In the context of numerical linear algebra, what is meant by 'ill-conditioning'?

Accepted Answer

A matrix is ill-conditioned if small changes in the input data can lead to large changes in the solution.

Answer

A matrix is ill-conditioned if its determinant is zero.

Question 39

The power iteration method is primarily used to find:

Accepted Answer

The dominant eigenvalue of a matrix

Answer

All eigenvalues of a matrix

Answer

The inverse of a matrix

Answer

The solution to a system of linear equations

Question 40

What is the primary purpose of using a preconditioner in an iterative method for solving linear systems?

Accepted Answer

To improve the convergence rate of the iterative method.

Answer

To reduce the computational cost of the method.

Answer

To increase the accuracy of the solution.

Question 41

Which of the following matrices is a diagonal matrix?

Accepted Answer

[2 0; 0 3]

Answer

[2 0; 3 4]

Answer

[2 3; 0 4]

Question 42

Which of the following is an eigenvalue of the matrix [1 2; 2 5]?

Accepted Answer

3

Answer

4

Answer

6

Answer

2

Question 43

What is the condition number of a matrix?

Accepted Answer

A measure of how sensitive a matrix is to perturbations

Answer

The number of eigenvalues of a matrix

Answer

The determinant of a matrix

Question 44

What is the complexity of Gaussian elimination for solving a system of n linear equations?

Accepted Answer

O(n^3)

Answer

O(n log n)

Answer

O(2^n)

Answer

O(n^2)

Question 45

Which of the following is a positive definite matrix?

Accepted Answer

A symmetric matrix with positive eigenvalues

Answer

A symmetric matrix with negative eigenvalues

Answer

A matrix with all positive entries

Question 46

Which of the following is a method for computing eigenvalues and eigenvectors?

Accepted Answer

Power iteration

Answer

Cramer's rule

Answer

Jacobi iteration

Answer

Gaussian elimination