GT Linear Algebra Module 3 True or False

If A is n × n, and A does not have n pivots, then det(A) = 0

True

If A is an n × n stochastic matrix and ~p is a column of A, then ~p is a probability
vector.

True

If A is n × n and upper triangular then the eigenvalues of A are non-zero.

Falso

f two matrices have the same eigenvalues, then the matrices are similar.

False

If A is n × n and not invertible, then A cannot be diagonalizable.

False

f an n × n matrix has n distinct eigenvalues, then the matrix is diagonalizable.

True

The steady-state of the Google matrix for any web with at least two pages is unique
when the damping factor, p, is equal to 0.85.

True

If A and B are n × n matrices, n > 1, and B is obtained by swapping two of the
rows of A, then det A = det B.

false

A stochastic matrix that is not regular can have a unique steady-state vector

true

If A is square and row equivalent to an identity matrix, then det(A) 6 = 0.

true

the steady state of a stochastic matrix is unique.

false

A 2 × 2 matrix A whose rank is 1 must have an eigenvalue that is equal to zero

true

If A is square and similar to the identity matrix, then A is the identity matrix.

true

f A is n × n and not diagonalizable, then A is not invertible

false

f A is n × n and diagonalizable, then A has n distinct eigenvalues.

false

f the characteristic polynomial of a 2 × 2 matrix has no real roots, then A must
have two complex eigenvalues.

true

f A and B are invertible n × n matrices, then det(AB) 6 = 0.

true

If an eigenvalue of n×n matrix A is λ = 1, then dim(Null(A−I)) = n−1.

false

f A is n × n, and there exists a ~b ∈ Rn such that A~x = ~b is inconsistent, then
det(A) = 0.

true

steady-state vector of a regular stochastic matrix P is unique.

true

f λ = 0 is an eigenvalue of A, then the matrix A is non-singular.

false

f matrices A and B are similar then A and B have the same eigenvalues.

true

If A is n × n and diagonalizable, then A is invertible.

false

uppose A is a 3 × 3 matrix with two eigenvalues, λ1 and λ2. If the geometric
multiplicity of λ1 is 1, and the geometric multiplicity of λ2 is 2, then A must be
diagonalizable.

true

f A is a real n × n matrix and n is odd, at least one of the eigenvalues of A is
real.

true

If det(A) = 4 and A is a 2 × 2 matrix, then det(2A) = 8.

false

f A is an n × n matrix and has eigenvector ~x, then 2~x is also an eigenvector of A.

true

Swapping the rows of A does not change the value of det(A)

false

ny stochastic matrix with a zero entry cannot be regular

false

n eigenspace is a subspace spanned by a single eigenvector.

false

non-zero matrix A can be similar to the zero matrix.

false

If A is n × n and not diagonalizable, then A is not invertible.

false

he n × n zero matrix can be diagonalized

true

The Google matrix for any web with at least two pages is always regular stochastic
when the damping factor, p, is equal to 0.85.

true

If A is n × n and invertible, then det(A3) 6 = 0.

true

If A is a square matrix, ~v and ~w are eigenvectors of A, then ~v + ~w is also an
eigenvector of A.

false

wapping the rows of A does not change the value of det(A)

false

f q is a steady-state vector for stochastic matrix P , then the Markov Chain xk+1 =
P xk converges to q as k → ∞

false

n eigenvalue of a matrix could be associated with two linearly independent eigen-
vectors.

true

f matrices A and B have the same eigenvalues, then A and B are similar.

false

uppose A is a 4 × 4 matrix that has exactly 2 distinct eigenvalues. If both of them
have geometric multiplicity 2, then A can be diagonalized.

true

If A is a real 2 × 2 singular matrix, then both eigenvalues of A cannot have an
imaginary component.

true

If det(A) = 3 and A is a 2 × 2 matrix, then det(2A) = 6.

false

If a stochastic matrix is not regular then it cannot have a steady state.

false

If A is a n×n, and det(A) = 3, then Ax = b has a solution for all b ∈ Rn

true

steady-state vector of a stochastic matrix P is unique.

false

n eigenspace of a square matrix A has a basis that consists of eigenvectors.

true

If A and B are 2 × 2 similar matrices, then A and B have the same rank.

true

uppose A is a diagonalizable n × n matrix, x and b are vectors in Rn. The linear
system Ax = b has a solution for all n.

false

f matrix A is n × n and has n distinct eigenvalues, then rankA = n.

false

A 2×2 matrix A with characteristic polynomial λ2 +1 has two complex eigenvalues

true

If A and B are n × n matrices, n > 1, and B is obtained by swapping two of the
rows of A, then det A = − det B.

true

1 is always an eigenvalue for any stochastic matrix P .

true

if E is a 2 × 2 elementary matrix, then det(E) = 1.

false

he dimension of an eigenspace of a square matrix A is one.

false

f A is a diagonalizable n × n matrix, then A has n distinct eigenvalues.

false

f an n × n matrix has n distinct eigenvalues, then Ax = b has a solution for all b.

false

A 2 × 2 matrix A with characteristic polynomial λ2 + 1 has two real eigenvalues.

false

f det(A) = 3 and A is a 3 × 3 matrix, then det(−A) = −3.

true

If an eigenvalue of n × n matrix A is λ = 2, then dim(Null(A − 2I)) = n − 1.

false

If A is a n × n, and det(A) = 3, then 3 is an eigenvalue of A.

false

A steady state vector of a regular stochastic matrix only has positive entries.

true

f A is an n × n matrix and A − 3I is non-singular, then 3 is an eigenvalue of A

false

f matrices A and B are similar then A and B have the same characteristic polynomial.

true

f A is a diagonalizable n×n matrix, then it is similar to a diagonal matrix.

true

he only 2 × 2 matrix that has the eigenvalues λ1 = λ2 = 0 is the zero matrix

false

f A ∈ R2×2 has complex eigenvalues λ1 and λ2, then |λ1| = |λ2|.

true

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

false

A regular stochastic matrix has full rank.

false

If E is n n × n elementary matrix, then det(E) 6 = 0.

true

he set of all probability vectors in Rn forms a subspace of Rn

false

If A is a n × n and A + 6I is singular, then −6 is an eigenvalue of the matrix.

true

If A is n × n and is similar to the zero matrix, then A must be equal to the zero
matrix.

true

f A is a diagonalizable n × n matrix, then rank(A) = n

false

If A is a square matrix that is similar to a diagonal matrix with distinct eigenvalues,
then A is diagonalizable.

true

If A is a 2×2 matrix and TA = Ax is a transform that rotates vectors by π/4 radians
clockwise about the origin.

true

The geometric multiplicity of an eigenvalue λ can be zero

false

If det(A) = 0, then zero is a root of the characteristic polynomial of A.

true

very stochastic matrix has at least one steady-state vector.

true

If X is an echelon form of n×n matrix Y , then X and Y have the same eigenvalues

false

If A is a square matrix that is similar to a diagonal matrix, then A must be a
diagonal matrix.

false

f a matrix A has n linearly independent eigenvectors, then A is diagonalizable

true

f a matrix A is diagonalizable, then there exists a matrix P and a diagonal matrix
D such that A = P DP −1 and A has the same eigenvalues as D.

true

A 2 × 2 matrix A with characteristic polynomial p(λ) = λ2 + 32 has two real
eigenvalues.

false

Swapping the columns of A does not change the value of det(A).

false

If ~v is an eigenvector of n × n matrices A and B, then ~v is an eigenvector of AB

true