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web.groovymark@gmail.com
- November 12, 2024
Question 21
Which of the following describes an antisymmetric relation?
a) If a is related to b and b is related to a, then a=b
b) If a is related to b and b is related to a, then a≠b
c) If a is related to b, then b is related to a
d) If a is related to b, then a is also related to c
Correct Answer: a) If a is related to b and b is related to a, then a=ba = ba=b
Explanation: An antisymmetric relation is one where if a is related to b and b is related to a, then a must equal b.
Question 22
Which of the following describes a transitive relation?
a) If a is related to b and b is related to c, then a is related to c
b) If a is related to b, then b is related to a
c) If a is related to b, then a is equal to b
d) If a is related to b, then a is not related to c
Correct Answer: a) If a is related to b and b is related to c, then a is related to c
Explanation: A transitive relation is one where if a is related to b and b is related to c, then a is also related to c.
Question 23
Which of the following is a property of an equivalence relation?
a) It is reflexive, symmetric, and transitive
b) It is reflexive and antisymmetric
c) It is antisymmetric and transitive
d) It is transitive and antisymmetric
Correct Answer: a) It is reflexive, symmetric, and transitive
Explanation: An equivalence relation is one that satisfies the properties of being reflexive, symmetric, and transitive.
Question 24
Which of the following describes a total order?
a) A binary relation where every pair of elements is comparable
b) A binary relation where no pair of elements is comparable
c) A binary relation where some pairs of elements are comparable
d) A binary relation that is reflexive but not transitive
Correct Answer: a) A binary relation where every pair of elements is comparable
Explanation: A total order is a binary relation where every pair of elements is comparable, meaning one element is always either less than or greater than the other.
Question 25
Which of the following describes a strict order?
a) A binary relation that is transitive and antisymmetric
b) A binary relation that is reflexive and symmetric
c) A binary relation that is transitive and symmetric
d) A binary relation that is reflexive and antisymmetric
Correct Answer: a) A binary relation that is transitive and antisymmetric
Explanation: A strict order is a binary relation that is transitive and antisymmetric, meaning no element is related to itself, and the relation is preserved through transitivity.
Question 26
What is a Hasse diagram used for?
a) To represent the order of elements in a partially ordered set
b) To represent the equivalence classes of a set
c) To represent the union of two sets
d) To represent the Cartesian product of two sets
Correct Answer: a) To represent the order of elements in a partially ordered set
Explanation: A Hasse diagram is a graphical representation of a partially ordered set, showing the order relations between the elements.
Question 27
What is the key property of a directed acyclic graph (DAG)?
a) It contains no cycles
b) It contains exactly one cycle
c) It contains multiple cycles
d) It contains parallel edges
Correct Answer: a) It contains no cycles
Explanation: A directed acyclic graph (DAG) is a graph with directed edges and no cycles, meaning there is no way to return to the starting vertex by following the edges.
Question 28
Which of the following describes a matrix that is invertible?
a) A square matrix that has an inverse
b) A rectangular matrix that has an inverse
c) A matrix where all elements are non-zero
d) A matrix that cannot be inverted
Correct Answer: a) A square matrix that has an inverse
Explanation: An invertible matrix is a square matrix that has an inverse, meaning it can be multiplied by another matrix to yield the identity matrix.
Question 29
Which of the following describes Gaussian elimination?
a) A method for solving systems of linear equations by reducing a matrix to row echelon form
b) A method for calculating the inverse of a matrix
c) A method for multiplying matrices
d) A method for transposing a matrix
Correct Answer: a) A method for solving systems of linear equations by reducing a matrix to row echelon form
Explanation: Gaussian elimination is a method used to solve systems of linear equations by reducing the coefficient matrix to row echelon form.
Question 30
What does the rank of a matrix represent?
a) The number of non-zero rows in its row echelon form
b) The number of rows in the matrix
c) The number of columns in the matrix
d) The number of zero rows in its row echelon form
Correct Answer: a) The number of non-zero rows in its row echelon form
Explanation: The rank of a matrix is the number of non-zero rows in its row echelon form, which corresponds to the dimension of the row space.
Question 31
What is an augmented matrix?
a) A matrix that includes an extra column representing the constants from a system of equations
b) A matrix where all elements are zero
c) A matrix that represents the inverse of another matrix
d) A matrix that has been transposed
Correct Answer: a) A matrix that includes an extra column representing the constants from a system of equations
Explanation: An augmented matrix is used in solving systems of linear equations and includes an extra column for the constants from the equations.
Question 32
Which of the following is true for an elementary row operation?
a) Swapping two rows of a matrix
b) Multiplying a row by a scalar
c) Adding a multiple of one row to another row
d) All of the above
Correct Answer: d) All of the above
Explanation: Elementary row operations include swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another.
Question 33
What does the determinant of a square matrix represent?
a) A scalar value that can indicate whether a matrix is invertible
b) The sum of the diagonal elements of the matrix
c) The product of all elements in the matrix
d) The number of non-zero rows in the matrix
Correct Answer: a) A scalar value that can indicate whether a matrix is invertible
Explanation: The determinant of a square matrix is a scalar value that can be used to determine whether the matrix is invertible. If the determinant is zero, the matrix is not invertible.
Question 34
Which of the following describes matrix transposition?
a) Swapping the rows and columns of a matrix
b) Swapping the elements on the main diagonal
c) Adding the elements of two matrices
d) Multiplying the elements of two matrices
Correct Answer: a) Swapping the rows and columns of a matrix
Explanation: Transposing a matrix means swapping its rows and columns, resulting in a new matrix where the element at position (i,j) in the original matrix becomes the element at (j,i).
Question 35
Which of the following is true for the inverse of a square matrix?
a) The product of the matrix and its inverse is the identity matrix
b) The inverse of a matrix is equal to the matrix itself
c) A matrix always has an inverse
d) The inverse of a matrix is found by multiplying all elements by -1
Correct Answer: a) The product of the matrix and its inverse is the identity matrix
Explanation: The inverse of a square matrix is such that when the matrix is multiplied by its inverse, the result is the identity matrix.
Question 36
Which of the following is a property of the identity matrix?
a) It has 1s on the diagonal and 0s elsewhere
b) It has 0s on the diagonal and 1s elsewhere
c) It has only 0s
d) It has only 1s
Correct Answer: a) It has 1s on the diagonal and 0s elsewhere
Explanation: The identity matrix is a square matrix with 1s on the diagonal and 0s in all other positions.
Question 37
What does the trace of a matrix represent?
a) The sum of the diagonal elements of the matrix
b) The product of the diagonal elements of the matrix
c) The sum of all elements in the matrix
d) The number of non-zero elements in the matrix
Correct Answer: a) The sum of the diagonal elements of the matrix
Explanation: The trace of a square matrix is the sum of its diagonal elements.
Question 38
Which of the following describes a diagonal matrix?
a) A matrix where all non-diagonal elements are zero
b) A matrix where all diagonal elements are zero
c) A matrix where all elements are non-zero
d) A matrix where all rows are equal
Correct Answer: a) A matrix where all non-diagonal elements are zero
Explanation: A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.
Question 39
Which of the following is true for the transpose of a matrix?
a) The transpose of the transpose is the original matrix
b) The transpose of a matrix is always the identity matrix
c) The transpose of a matrix is always a diagonal matrix
d) The transpose of a matrix is always its inverse
Correct Answer: a) The transpose of the transpose is the original matrix
Explanation: Transposing a matrix twice returns the original matrix.
Question 40
What does Gaussian elimination aim to achieve?
a) Reducing a matrix to row echelon form
b) Finding the determinant of a matrix
c) Transposing a matrix
d) Finding the inverse of a matrix
Correct Answer: a) Reducing a matrix to row echelon form
Explanation: Gaussian elimination is a process used to reduce a matrix to row echelon form, simplifying the process of solving systems of linear equations.
