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

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

true

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

false

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

true

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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

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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

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

true

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

true

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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

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"Every matrix has a unique row echelon form"

false

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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

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

true

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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

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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

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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

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

false

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

false

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

false

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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

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

true

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

false

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

true

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

true

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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

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

false

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

true

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

true

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

false

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

false

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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

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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

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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

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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

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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

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

true

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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

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

true

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

false

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

false

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"Every elementary matrix is invertible"

true

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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

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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

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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

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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

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

false

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

true

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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

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

true

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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

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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

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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

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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

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

true

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

false

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

false

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

true

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

true

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

false

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

false

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

true

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

true

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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

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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

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

false

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

true

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

false

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

false

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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

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

false

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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

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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

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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

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

false

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

true

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

true

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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

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

true

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

false

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

false

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

false

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

true

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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

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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

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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

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

true

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