LinAlg T/F

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166 Terms

1
New cards
"A linear system whose equations are all homogeneous must be consistent"
true
2
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"Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation"
false
3
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"A single linear equation with two or more unknowns must have infinitely many solutions"
true
4
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"If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent"
false
5
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"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
6
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"Elementary row operations permit one row of an augmented matrix to be subtracted from another"
true
7
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"If a matrix is in reduced row echelon form, then it is also in row echelon form"
true
8
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"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
9
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"Every matrix has a unique row echelon form"
false
10
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"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
11
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"All leading 1's in a matrix in row echelon form must occur in different columns"
true
12
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"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
13
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"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
14
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"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
15
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"If a linear system has more unknowns than equations, then it must have infinitely many solutions"
false
16
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"An m × n matrix has m column vectors and n row vectors."
false
17
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"If A and B are 2 × 2 matrices, then AB \= BA."
false
18
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"The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B"
false
19
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"For every matrix A, it is true that (AT)^T \= A"
true
20
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"If A and B are square matrices of the same order, then (AB)^T \= A^T B^T"
false
21
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"For every square matrix A, it is true that tr(A^T) \= tr(A)"
true
22
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"If A is an n × n matrix and c is a scalar, then tr(cA) \= c tr(A)"
true
23
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"If A, B, and C are matrices of the same size such that A − C \= B − C, then A \= B"
true
24
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"If A, B, and C are square matrices of the same order such that AC \= BC, then A \= B"
false
25
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"If AB + BA is defined, then A and B are square matrices of the same size"
true
26
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"If B has a column of zeros, then so does AB if this product is defined."
true
27
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"If B has a column of zeros, then so does BA if this product is defined"
false
28
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"Two n × n matrices, A and B, are inverses of one another if and only if AB \= BA \= 0"
false
29
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"For all square matrices A and B of the same size, it is true that (A + B)^2 \= A^2 + 2AB + B^2"
false
30
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"For all square matrices A and B of the same size, it is true that A^2 − B^2 \= (A − B)(A + B)"
false
31
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If A and B are invertible matrices of the same size, then AB is invertible and (AB)^−1 \= A^−1 B^−1"
false
32
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"If A and B are matrices such that AB is defined, then it is true that (AB)^T \= A^T B^T"
false
33
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"If A and B are matrices of the same size and k is a constant, then (kA + B)^T \= kA^T + B^T"
true
34
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"If A is an invertible matrix, then so is A^T"
true
35
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"f p(x) \= a0 + a1x + a2x^2 + ⋯ + amx^m and I is an identity matrix, then p(I) \= a0 + a1 + a2 + ⋯ + am"
false
36
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"A square matrix containing a row or column of zeros cannot be invertible"
true
37
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"The sum of two invertible matrices of the same size must be invertible"
false
38
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"The product of two elementary matrices of the same size must be an elementary matrix"
false
39
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"Every elementary matrix is invertible"
true
40
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"If A and B are row equivalent, and if B and C are row equivalent, then A and C are row equivalent"
true
41
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"If A is an n × n matrix that is not invertible, then the linear system Ax \= 0 has infinitely many solutions"
true
42
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"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
43
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"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
44
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"An expression of an invertible matrix A as a product of elementary matrices is unique"
false
45
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"It is impossible for a system of linear equations to have exactly two solutions"
true
46
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"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
47
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"If A and B are n × n matrices such that AB \= I, then BA \= I"
true
48
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"If A and B are row equivalent matrices, then the linear systems Ax \= 0 and Bx \= 0 have the same solution set"
true
49
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"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
50
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"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
51
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"Let A and B be n × n matrices. If A or B (or both) are not invertible, then neither is AB"
true
52
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"The transpose of a diagonal matrix is a diagonal matrix."
true
53
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"The transpose of an upper triangular matrix is an upper triangular matrix"
false
54
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"the sum of an upper triangular matrix and a lower triangular matrix is a diagonal matrix"
false
55
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"All entries of a symmetric matrix are determined by the entries occurring on and above the main diagonal"
true
56
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"All entries of an upper triangular matrix are determined by the entries occurring on and above the main diagonal."
true
57
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"The inverse of an invertible lower triangular matrix is an upper triangular matrix"
false
58
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"A diagonal matrix is invertible if and only if all of its diagonal entries are positive"
false
59
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"The sum of a diagonal matrix and a lower triangular matrix is a lower triangular matrix"
true
60
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"A matrix that is both symmetric and upper triangular must be a diagonal matrix"
true
61
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"If A and B are n × n matrices such that A + B is symmetric, then A and B are symmetric"
false
62
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"If A and B are n × n matrices such that A + B is upper triangular, then A and B are upper triangular"
false
63
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"If A^2 is a symmetric matrix, then A is a symmetric matrix"
false
64
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"If kA is a symmetric matrix for some k ≠ 0, then A is a symmetric matrix"
true
65
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"If A is a 2 × 3 matrix, then the domain of the transformation TA is R^2"
false
66
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"If A is an m × n matrix, then the codomain of the transformation TA is R^n"
false
67
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"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
68
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"There are linear transformations from R^n to R^m that are not matrix transformations"
false
69
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"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
70
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"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
71
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"If b is a nonzero vector in R^n, then T(x) \= x + b is a matrix operator on R^n."
false
72
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"Two square matrices that have the same determinant must have the same size"
false
73
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"The minor Mij is the same as the cofactor Cij if i + j is even"
true
74
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"If A is a 3 × 3 symmetric matrix, then Cij \= Cji for all i and j"
true
75
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"The number obtained by a cofactor expansion of a matrix A is independent of the row or column chosen for the expansion"
true
76
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"If A is a square matrix whose minors are all zero, then det(A) \= 0"
true
77
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"The determinant of a lower triangular matrix is the sum of the entries along the main diagonal"
false
78
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"For every square matrix A and every scalar c, it is true that det (cA) \= c det(A)"
false
79
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"For all square matrices A and B, it is true that det(A + B) \= det(A) + det(B)"
false
80
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"For every 2 × 2 matrix A it is true that det(A^2) \= (det(A))^2"
true
81
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"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
82
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"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
83
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"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
84
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"If A is a square matrix with two identical columns, then det(A) \= 0"
true
85
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"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
86
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"If A is a 3 × 3 matrix, then det(2A) \= 2 det(A)"
false
87
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"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
88
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"If A and B are square matrices of the same size and A is invertible, then det(A^−1BA)\=det(B) "
true
89
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"A square matrix A is invertible if and only if det(A) \= 0"
false
90
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"The matrix of cofactors of A is precisely [adj(A)]^T"
true
91
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"For every n × n matrix A, we have A · adj(A)\=(det(A))I "
true
92
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"If A is a square matrix and the linear system Ax \= 0 has multiple solutions for x, then det(A) \= 0"
true
93
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"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
94
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"If E is an elementary matrix, then Ex \= 0 has only the trivial solution"
true
95
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"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
96
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"If A is invertible, then adj(A) must also be invertible"
true
97
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"If A has a row of zeros, then so does adj(A)"
false
98
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"A vector is any element of a vector space"
true
99
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"A vector space must contain at least two vectors"
false
100
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"If u is a vector and k is a scalar such that ku \= 0, then it must be true that k \= 0"
false