Studied by 0 People

0.0(0)

Get a hint

hint

Looks like no one added any tags here yet for you.

New cards166

Still learning0

Almost done0

Mastered0

166 Terms

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

"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

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

"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

New cards

New cards

"Every matrix has a unique row echelon form"

false

New cards

New cards

"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

New cards

New cards

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

true

New cards

New cards

"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

New cards

New cards

"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

New cards

New cards

"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

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

"Every elementary matrix is invertible"

true

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

"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

New cards

New cards

"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

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

"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

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

"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

New cards

New cards

"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

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

"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

New cards

New cards

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

false

New cards

New cards

"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

New cards

New cards

"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

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

true

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

false

New cards

New cards

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

true

New cards

New cards

"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

New cards

New cards

"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

New cards

New cards

"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

New cards

New cards

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

true

New cards