-
web.groovymark@gmail.com
- November 12, 2024
Question 41
Which of the following describes a matrix in row echelon form?
a) Each leading entry is 1, and all elements above and below the leading entries are zero
b) Each row contains only zero elements
c) The matrix is symmetric
d) The matrix has no non-zero rows
Correct Answer: a) Each leading entry is 1, and all elements above and below the leading entries are zero
Explanation: In row echelon form, the leading entry of each row is 1, and the entries above and below the leading entries are zero.
Question 42
Which of the following describes a non-invertible matrix?
a) A matrix with a determinant of zero
b) A matrix with all elements equal to zero
c) A matrix with non-zero rows
d) A matrix with no diagonal elements
Correct Answer: a) A matrix with a determinant of zero
Explanation: A matrix is non-invertible, or singular, if its determinant is zero.
Question 43
Which of the following describes an elementary matrix?
a) A matrix that results from performing a single elementary row operation on the identity matrix
b) A matrix with all elements equal to zero
c) A matrix that is symmetric
d) A matrix that has no inverse
Correct Answer: a) A matrix that results from performing a single elementary row operation on the identity matrix
Explanation: An elementary matrix is obtained by performing a single elementary row operation on the identity matrix.
Question 44
Which of the following best describes a symmetric matrix?
a) A matrix that is equal to its transpose
b) A matrix where all elements are non-zero
c) A matrix where all elements are zero
d) A matrix with equal row and column sums
Correct Answer: a) A matrix that is equal to its transpose
Explanation: A symmetric matrix is one where the transpose is equal to the original matrix, meaning A=AT.
Question 45
Which of the following describes an orthogonal matrix?
a) A matrix where the transpose is equal to the inverse
b) A matrix where the rows are equal to the columns
c) A matrix where all elements are positive
d) A matrix where all elements are negative
Correct Answer: a) A matrix where the transpose is equal to the inverse
Explanation: An orthogonal matrix is one where the transpose is equal to the inverse, meaning AT=A−1.
Question 46
Which of the following describes the inverse of a matrix?
a) The matrix that, when multiplied by the original matrix, gives the identity matrix
b) The matrix that, when added to the original matrix, gives the zero matrix
c) The matrix that has all elements equal to zero
d) The matrix that has all elements equal to one
Correct Answer: a) The matrix that, when multiplied by the original matrix, gives the identity matrix
Explanation: The inverse of a matrix is the matrix that, when multiplied by the original matrix, results in the identity matrix.
Question 47
What does the identity matrix represent in matrix multiplication?
a) It acts as a multiplicative identity, meaning multiplying any matrix by the identity matrix leaves the matrix unchanged
b) It acts as an additive identity, meaning adding the identity matrix to any matrix leaves the matrix unchanged
c) It acts as a zero matrix, meaning multiplying any matrix by the identity matrix results in a zero matrix
d) It acts as a diagonal matrix, meaning multiplying any matrix by the identity matrix results in a diagonal matrix
Correct Answer: a) It acts as a multiplicative identity, meaning multiplying any matrix by the identity matrix leaves the matrix unchanged
Explanation: The identity matrix is the multiplicative identity for matrices, meaning multiplying any matrix by the identity matrix leaves the matrix unchanged.
Question 48
What is the primary purpose of performing matrix row reduction?
a) To simplify the process of solving systems of linear equations
b) To find the determinant of a matrix
c) To find the inverse of a matrix
d) To transpose a matrix
Correct Answer: a) To simplify the process of solving systems of linear equations
Explanation: Matrix row reduction simplifies the process of solving systems of linear equations by transforming the coefficient matrix into a simpler form, such as row echelon form.
Question 49
Which of the following is true for a square matrix?
a) The number of rows is equal to the number of columns
b) The number of rows is greater than the number of columns
c) The number of columns is greater than the number of rows
d) The matrix has no diagonal elements
Correct Answer: a) The number of rows is equal to the number of columns
Explanation: A square matrix has the same number of rows and columns, forming an n×n matrix.
Question 50
Which of the following describes a unitary matrix?
a) A matrix whose inverse is equal to its conjugate transpose
b) A matrix where all elements are zero
c) A matrix where all elements are one
d) A matrix where the determinant is zero
Correct Answer: a) A matrix whose inverse is equal to its conjugate transpose
Explanation: A unitary matrix is one where the inverse is equal to the conjugate transpose, meaning U−1=U∗.
