Final T/F Study Guide (Chapters 1-4

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

The linear system x − y = 3 2x − 2y = k cannot have a unique solution, regardless of the value of k.

True

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

The linear system with corresponding augmented matrix [ 2 −1 4 0 0 −1 ] is consistent.

False

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

The matrix [ 1 2 3 4 5 6] has no main diagonal.

True

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 tr(𝐴𝐵) = tr(𝐴)tr(𝐵)

False

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

False

For every square matrix 𝐴, it is true that tr(𝐴𝑇) = tr(𝐴).

True

If 𝐴 is a 6 × 4 matrix and 𝐵 is an m × n matrix such that 𝐵 𝑇𝐴𝑇 is a 2 × 6 matrix, then m = 4 and n = 2.

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 = 𝐴2 + 2𝐴𝐵 + 𝐵2 .

False

For all square matrices 𝐴 and 𝐵 of the same size, it is true that 𝐴2 − 𝐵2 = (𝐴 − 𝐵)(𝐴 + 𝐵).

False

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

False

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

False

The matrix 𝐴 = [ a b c d] is invertible if and only if ad − bc ≠ 0

True

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

True

If 𝐴 is an invertible matrix, then so is 𝐴𝑇.

True

If p(x) = a0 + a1 x + a2 x 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 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 = 𝐼n , then 𝐵𝐴 = 𝐼n .

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 the system (𝑆−1𝐴𝑆)x = b, then 𝑆x is a solution to the 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 𝐴2 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 ∶ 𝑅n → 𝑅m 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 ∶ 𝑅n → 𝑅n 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 ∶ 𝑅n → 𝑅m 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

The determinant of the 2 × 2 matrix [ a b c d] is ad + bc.

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(𝐴2 ) = (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 an n × n matrix and 𝐵 is obtained from 𝐴 by multiplying each row of 𝐴 by its row number, then det(𝐵) = n(n + 1) 2 det(𝐴)

False

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

True

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

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

False

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

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

True

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

False

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

True

For every n × n matrix 𝐴, we have 𝐴 ⋅ adj(𝐴) = (det(𝐴))𝐼n

True

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

True

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 𝐼n .

True