LinAlg T/F

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hint

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

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

true

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

false

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

true

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

false

"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

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

true

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

true

"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

"Every matrix has a unique row echelon form"

false

"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

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

true

"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

"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

"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

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

false

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

false

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

false

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

false

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

true

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

false

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

true

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

true

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

true

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

false

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

true

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

true

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

false

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

false

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

false

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

false

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

false

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

false

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

true

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

true

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

false

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

true

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

false

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

false

"Every elementary matrix is invertible"

true

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

true

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

true

"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

"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

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

false

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

true

"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

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

true

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

true

"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

"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

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

true

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

true

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

false

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

false

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

true

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

true

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

false

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

false

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

true

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

true

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

false

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

false

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

false

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

true

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

false

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

false

"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

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

false

"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

"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

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

false

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

false

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

true

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

true

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

true

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

true

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

false

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

false

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

false

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

true

"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

"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

"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

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

true