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1

"A linear system whose equations are all homogeneous must be consistent"

true

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2

"Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation"

false

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3

"A single linear equation with two or more unknowns must have infinitely many solutions"

true

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4

"If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent"

false

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5

"If each equation in a consistent linear system is multiplied through by a constant c, then all solutions to the new system can be obtained by multiplying solutions from the original system by c."

false

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6

"Elementary row operations permit one row of an augmented matrix to be subtracted from another"

true

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7

"If a matrix is in reduced row echelon form, then it is also in row echelon form"

true

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8

"If an elementary row operation is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form"

false

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9

"Every matrix has a unique row echelon form"

false

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10

"A homogeneous linear system in n unknowns whose corresponding augmented matrix has a reduced row echelon form with r leading 1's has n − r free variables"

true

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11

"All leading 1's in a matrix in row echelon form must occur in different columns"

true

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12

"If every column of a matrix in row echelon form has a leading 1, then all entries that are not leading 1's are zero"

false

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13

"If a homogeneous linear system of n equations in n unknowns has a corresponding augmented matrix with a reduced row echelon form containing n leading 1's, then the linear system has only the trivial solution"

true

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14

"If the reduced row echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions"

false

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15

"If a linear system has more unknowns than equations, then it must have infinitely many solutions"

false

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16

"An m × n matrix has m column vectors and n row vectors."

false

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17

"If A and B are 2 × 2 matrices, then AB = BA."

false

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18

"The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B"

false

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19

"For every matrix A, it is true that (AT)^T = A"

true

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20

"If A and B are square matrices of the same order, then (AB)^T = A^T B^T"

false

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21

"For every square matrix A, it is true that tr(A^T) = tr(A)"

true

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22

"If A is an n × n matrix and c is a scalar, then tr(cA) = c tr(A)"

true

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23

"If A, B, and C are matrices of the same size such that A − C = B − C, then A = B"

true

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24

"If A, B, and C are square matrices of the same order such that AC = BC, then A = B"

false

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25

"If AB + BA is defined, then A and B are square matrices of the same size"

true

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26

"If B has a column of zeros, then so does AB if this product is defined."

true

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27

"If B has a column of zeros, then so does BA if this product is defined"

false

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28

"Two n × n matrices, A and B, are inverses of one another if and only if AB = BA = 0"

false

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29

"For all square matrices A and B of the same size, it is true that (A + B)^2 = A^2 + 2AB + B^2"

false

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30

"For all square matrices A and B of the same size, it is true that A^2 − B^2 = (A − B)(A + B)"

false

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31

If A and B are invertible matrices of the same size, then AB is invertible and (AB)^−1 = A^−1 B^−1"

false

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32

"If A and B are matrices such that AB is defined, then it is true that (AB)^T = A^T B^T"

false

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33

"If A and B are matrices of the same size and k is a constant, then (kA + B)^T = kA^T + B^T"

true

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34

"If A is an invertible matrix, then so is A^T"

true

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35

"f p(x) = a0 + a1x + a2x^2 + ⋯ + amx^m and I is an identity matrix, then p(I) = a0 + a1 + a2 + ⋯ + am"

false

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36

"A square matrix containing a row or column of zeros cannot be invertible"

true

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37

"The sum of two invertible matrices of the same size must be invertible"

false

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38

"The product of two elementary matrices of the same size must be an elementary matrix"

false

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39

"Every elementary matrix is invertible"

true

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40

"If A and B are row equivalent, and if B and C are row equivalent, then A and C are row equivalent"

true

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41

"If A is an n × n matrix that is not invertible, then the linear system Ax = 0 has infinitely many solutions"

true

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42

"If A is an n × n matrix that is not invertible, then the matrix obtained by interchanging two rows of A cannot be invertible"

true

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43

"If A is invertible and a multiple of the first row of A is added to the second row, then the resulting matrix is invertible"

true

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44

"An expression of an invertible matrix A as a product of elementary matrices is unique"

false

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45

"It is impossible for a system of linear equations to have exactly two solutions"

true

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46

"If A is a square matrix, and if the linear system Ax = b has a unique solution, then the linear system Ax = c also must have a unique solution"

true

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47

"If A and B are n × n matrices such that AB = I, then BA = I"

true

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48

"If A and B are row equivalent matrices, then the linear systems Ax = 0 and Bx = 0 have the same solution set"

true

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49

"Let A be an n × n matrix and S is an n × n invertible matrix. If x is a solution to system (S −1 AS)x = b, then Sx is a solution system Ay = Sb"

true

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50

"Let A be an n × n matrix. The linear system Ax = 4x has a unique solution if and only if A − 4I is an invertible matrix"

true

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51

"Let A and B be n × n matrices. If A or B (or both) are not invertible, then neither is AB"

true

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52

"The transpose of a diagonal matrix is a diagonal matrix."

true

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53

"The transpose of an upper triangular matrix is an upper triangular matrix"

false

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54

"the sum of an upper triangular matrix and a lower triangular matrix is a diagonal matrix"

false

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55

"All entries of a symmetric matrix are determined by the entries occurring on and above the main diagonal"

true

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56

"All entries of an upper triangular matrix are determined by the entries occurring on and above the main diagonal."

true

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57

"The inverse of an invertible lower triangular matrix is an upper triangular matrix"

false

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58

"A diagonal matrix is invertible if and only if all of its diagonal entries are positive"

false

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59

"The sum of a diagonal matrix and a lower triangular matrix is a lower triangular matrix"

true

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60

"A matrix that is both symmetric and upper triangular must be a diagonal matrix"

true

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61

"If A and B are n × n matrices such that A + B is symmetric, then A and B are symmetric"

false

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62

"If A and B are n × n matrices such that A + B is upper triangular, then A and B are upper triangular"

false

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63

"If A^2 is a symmetric matrix, then A is a symmetric matrix"

false

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64

"If kA is a symmetric matrix for some k ≠ 0, then A is a symmetric matrix"

true

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65

"If A is a 2 × 3 matrix, then the domain of the transformation TA is R^2"

false

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66

"If A is an m × n matrix, then the codomain of the transformation TA is R^n"

false

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67

"There is at least one linear transformation T : R^n → R^m for which T(2x) = 4T(x) for some vector x in R^n"

true

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68

"There are linear transformations from R^n to R^m that are not matrix transformations"

false

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69

"If TA : R^n → R^n and if TA(x) = 0 for every vector x in R^n, then A is the n × n zero matrix"

true

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70

"There is only one matrix transformation T : R^n → R^m such that T(− x) = − T(x) for every vector x in R^n"

false

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71

"If b is a nonzero vector in R^n, then T(x) = x + b is a matrix operator on R^n."

false

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72

"Two square matrices that have the same determinant must have the same size"

false

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73

"The minor Mij is the same as the cofactor Cij if i + j is even"

true

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74

"If A is a 3 × 3 symmetric matrix, then Cij = Cji for all i and j"

true

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75

"The number obtained by a cofactor expansion of a matrix A is independent of the row or column chosen for the expansion"

true

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76

"If A is a square matrix whose minors are all zero, then det(A) = 0"

true

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77

"The determinant of a lower triangular matrix is the sum of the entries along the main diagonal"

false

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78

"For every square matrix A and every scalar c, it is true that det (cA) = c det(A)"

false

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79

"For all square matrices A and B, it is true that det(A + B) = det(A) + det(B)"

false

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80

"For every 2 × 2 matrix A it is true that det(A^2) = (det(A))^2"

true

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81

"If A is a 4 × 4 matrix and B is obtained from A by interchanging the first two rows and then interchanging the last two rows, then det(B) = det(A)"

true

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82

"If A is a 3 × 3 matrix and B is obtained from A by multiplying the first column by 4 and multiplying the third column by 3/4, then det(B) = 3 det(A)"

true

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83

"If A is a 3 × 3 matrix and B is obtained from A by adding 5 times the first row to each of the second and third rows, then det(B) = 25 det(A)"

false

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84

"If A is a square matrix with two identical columns, then det(A) = 0"

true

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85

"If the sum of the second and fourth row vectors of a 6 × 6 matrix A is equal to the last row vector, then det(A) = 0"

true

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86

"If A is a 3 × 3 matrix, then det(2A) = 2 det(A)"

false

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87

"If A and B are square matrices of the same size such that det(A) = det(B), then det(A + B) = 2 det(A)"

false

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88

"If A and B are square matrices of the same size and A is invertible, then det(A^−1BA)=det(B) "

true

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89

"A square matrix A is invertible if and only if det(A) = 0"

false

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90

"The matrix of cofactors of A is precisely [adj(A)]^T"

true

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91

"For every n × n matrix A, we have A · adj(A)=(det(A))I "

true

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92

"If A is a square matrix and the linear system Ax = 0 has multiple solutions for x, then det(A) = 0"

true

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93

"If A is an n × n matrix and there exists an n × 1 matrix b such that the linear system Ax = b has no solutions, then the reduced row echelon form of A cannot be I"

true

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94

"If E is an elementary matrix, then Ex = 0 has only the trivial solution"

true

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95

"If A is an invertible matrix, then the linear system Ax = 0 has only the trivial solution if and only if the linear system A^−1x = 0 has only the trivial solution"

true

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96

"If A is invertible, then adj(A) must also be invertible"

true

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97

"If A has a row of zeros, then so does adj(A)"

false

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98

"A vector is any element of a vector space"

true

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99

"A vector space must contain at least two vectors"

false

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100

"If u is a vector and k is a scalar such that ku = 0, then it must be true that k = 0"

false

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