A singular matrix is one that does not have an inverse, which occurs when the determinant of the matrix is zero.
2. What is the result of multiplying a matrix by its inverse?
a) The zero matrix
b) The original matrix
c) A diagonal matrix
d) The identity matrix
Answer:
d) The identity matrix
Explanation:
Multiplying a matrix by its inverse results in the identity matrix.
3. What represents the number of linearly independent rows or columns in a matrix?
a) Determinant
b) Trace
c) Rank
d) Eigenvalue
Answer:
c) Rank
Explanation:
The rank of a matrix is defined as the maximum number of linearly independent row vectors (or column vectors) in the matrix.
4. In a vector space, what are eigenvectors and eigenvalues associated with?
a) Matrix addition
b) Matrix inversion
c) A linear transformation
d) Vector cross product
Answer:
c) A linear transformation
Explanation:
Eigenvectors and eigenvalues are concepts associated with a linear transformation represented by a matrix. They represent directions in which the transformation only scales the vectors.
5. What is the primary operation performed in Gaussian elimination?
a) Multiplying matrices
b) Finding determinants
c) Row reduction
d) Calculating eigenvalues
Answer:
c) Row reduction
Explanation:
Gaussian elimination involves the use of elementary row operations to reduce a matrix to row echelon form or reduced row echelon form.
6. What is the trace of a matrix?
a) Sum of the elements along the main diagonal
b) Product of the elements along the main diagonal
c) Sum of the eigenvalues
d) Product of the eigenvalues
Answer:
a) Sum of the elements along the main diagonal
Explanation:
The trace of a matrix is defined as the sum of the elements on the main diagonal of the matrix.
7. In linear algebra, what is a diagonal matrix?
a) A matrix where all elements off the main diagonal are zero
b) A matrix with equal rows and columns
c) A symmetric matrix
d) A matrix with all elements equal
Answer:
a) A matrix where all elements off the main diagonal are zero
Explanation:
A diagonal matrix is a type of matrix where all entries outside the main diagonal are zero.
8. What is the determinant of a triangular matrix?
a) The sum of the diagonal elements
b) The product of the diagonal elements
c) Zero
d) One
Answer:
b) The product of the diagonal elements
Explanation:
The determinant of a triangular matrix (upper or lower) is the product of the elements on the main diagonal.
9. Which of the following is a property of the transpose of a matrix?
a) The transpose of a matrix is always singular
b) The transpose of a matrix changes its determinant
c) The transpose of a symmetric matrix is the same as the original matrix
d) The transpose of a matrix doubles its dimensions
Answer:
c) The transpose of a symmetric matrix is the same as the original matrix
Explanation:
The transpose of a matrix is achieved by flipping it over its diagonal, so the transpose of a symmetric matrix is the same as the original.
10. What does the vector cross product result in?
a) A scalar
b) A matrix
c) Another vector
d) A tuple
Answer:
c) Another vector
Explanation:
The cross product of two vectors results in a vector that is perpendicular to both of the original vectors.
11. What is a 'linear combination' of vectors?
a) The sum of vectors
b) The product of vectors
c) The sum of scalar multiples of vectors
d) The dot product of vectors
Answer:
c) The sum of scalar multiples of vectors
Explanation:
A linear combination involves adding together scalar multiples of vectors.
12. What does orthogonality mean in the context of vectors?
a) The vectors are parallel
b) The vectors are at a 45-degree angle to each other
c) The vectors are perpendicular to each other
d) The vectors are identical
Answer:
c) The vectors are perpendicular to each other
Explanation:
Orthogonality refers to the concept of two vectors being perpendicular, i.e., their dot product is zero.
13. What is a subspace in linear algebra?
a) A set of vectors including their linear combinations
b) The set of all possible matrices
c) A single point in a vector space
d) A line in a matrix
Answer:
a) A set of vectors including their linear combinations
Explanation:
A subspace is a subset of a vector space that is also a vector space, containing the zero vector and closed under vector addition and scalar multiplication.
14. What is the geometric interpretation of a determinant of a 2×2 matrix?
a) The length of the vectors
b) The angle between the vectors
c) The area of the parallelogram spanned by the column vectors
d) The volume of the space enclosed by the matrix
Answer:
c) The area of the parallelogram spanned by the column vectors
Explanation:
The absolute value of the determinant of a 2×2 matrix represents the area of the parallelogram formed by the column vectors of the matrix.
15. How many solutions does a system of linear equations have if its coefficient matrix is singular?
a) Exactly one solution
b) No solution
c) Either no solution or infinitely many solutions
d) Exactly two solutions
Answer:
c) Either no solution or infinitely many solutions
Explanation:
If the coefficient matrix of a system of linear equations is singular, the system has either no solution or infinitely many solutions.
16. What is the rank of the identity matrix?
a) Zero
b) One
c) The number of rows in the matrix
d) The number of columns in the matrix
Answer:
c) The number of rows in the matrix
Explanation:
The rank of the identity matrix is equal to the number of rows (or columns), as it is a full-rank matrix.
17. What is a 'vector space'?
a) A collection of vectors and matrices
b) A set of vectors that are closed under vector addition and scalar multiplication
c) A space where vectors are linearly independent
d) A graphical representation of vectors
Answer:
b) A set of vectors that are closed under vector addition and scalar multiplication
Explanation:
A vector space is a set of vectors along with two operations (vector addition and scalar multiplication) that satisfy certain axioms.
18. What does it mean for two matrices to be 'conformable for multiplication'?
a) They have the same number of rows
b) They have the same number of columns
c) The number of columns of the first matrix is equal to the number of rows of the second
d) Both matrices are square matrices
Answer:
c) The number of columns of the first matrix is equal to the number of rows of the second
Explanation:
For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.
19. What is the result of the dot product of two orthogonal vectors?
a) 1
b) 0
c) The sum of their magnitudes
d) The product of their magnitudes
Answer:
b) 0
Explanation:
The dot product of two orthogonal vectors is zero.
20. What is a 'basis' of a vector space?
a) A set of vectors that span the vector space
b) Any set of vectors in the vector space
c) A set of linearly dependent vectors
d) A set of vectors that span the vector space and are linearly independent
Answer:
d) A set of vectors that span the vector space and are linearly independent
Explanation:
A basis of a vector space is a set of linearly independent vectors that span the entire vector space.
21. What is the main purpose of LU decomposition of a matrix?
a) To simplify matrix addition
b) To find the inverse of a matrix
c) To solve systems of linear equations
d) To compute the determinant
Answer:
c) To solve systems of linear equations
Explanation:
LU decomposition decomposes a matrix into a product of a lower triangular matrix and an upper triangular matrix and is primarily used for solving systems of linear equations.
22. How is the length of a vector typically represented?
a) The sum of its components
b) The square root of the sum of the squares of its components
c) The maximum of its components
d) The product of its components
Answer:
b) The square root of the sum of the squares of its components
Explanation:
The length or magnitude of a vector in Euclidean space is given by the square root of the sum of the squares of its components.
23. What is a null space of a matrix?
a) The set of all vectors that the matrix maps to zero
b) The space spanned by the row vectors
c) The set of all non-zero vectors
d) The space spanned by the column vectors
Answer:
a) The set of all vectors that the matrix maps to zero
Explanation:
The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector.
24. What is the result of a matrix multiplied by the zero vector?
a) The identity vector
b) The zero vector
c) The original matrix
d) A random vector
Answer:
b) The zero vector
Explanation:
Multiplying any matrix by the zero vector always results in the zero vector.
25. What does 'linear independence' of a set of vectors imply?
a) The vectors can be expressed as a linear combination of each other
b) No vector in the set can be written as a linear combination of the others
c) The vectors are orthogonal to each other
d) The vectors form a basis for a vector space
Answer:
b) No vector in the set can be written as a linear combination of the others
Explanation:
A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others. This implies that the vectors do not overlap in terms of their span in the vector space.