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Linear Algebra

What is the definition of a matrix?
A mathematical operation
A rectangular array of numbers
A type of equation
A geometric shape
Which of the following is not a basic operation on matrices?
Addition
Subtraction
Multiplication
Division
What is the determinant of a 2x2 matrix?
The sum of the diagonal elements
The product of the diagonal elements
The difference between the diagonal elements
The determinant formula does not apply to 2x2 matrices
Which of the following is a vector space?
Set of all real numbers
Set of all positive integers
Set of all polynomials of degree 2 or less
Set of all even numbers
Set of all rational numbers
What is the rank of a matrix?
The number of rows
The number of columns
The sum of all elements
The maximum number of linearly independent rows or columns
Which of the following is an eigenvalue?
A scalar value that satisfies the equation Ax = ?x
A vector that satisfies the equation Ax = ?x
A determinant of a matrix
A coefficient in a polynomial equation
What does it mean for a matrix to be invertible?
The matrix has no inverse
The matrix can be multiplied by its inverse to obtain the identity matrix
The matrix has only one row
The matrix has only one column
What is the dot product of two vectors?
The sum of the products of their corresponding components
The difference between their magnitudes
The cross product of the vectors
The dot product is not defined for vectors
Which of the following is a valid matrix operation?
Transposition
Determinant
Inversion
Differentiation
Integration
What is the characteristic polynomial of a matrix?
A polynomial equation with the matrix as a variable
A polynomial equation with the eigenvalues as variables
A polynomial equation with the determinants as variables
The determinant of the matrix
Which of the following is true about positive definite matrices?
All eigenvalues are positive
All elements are positive
The determinant is positive
The matrix is symmetric
The matrix is invertible
What is the purpose of the singular value decomposition (SVD)?
To find the eigenvalues of a matrix
To factorize a matrix into three separate matrices
To calculate the determinant of a matrix
To solve systems of linear equations
What is the main application of linear algebra in Fourier transforms?
Image recognition
Audio signal processing
Data encryption
Frequency analysis
Which of the following is an example of a Markov process?
Weather forecasting
Stock market prediction
Random walk on a graph
Solving differential equations
Sorting algorithms
How does linear algebra contribute to the stability analysis of differential equations?
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