Linear Algebra Exam 3 T/F practice

A is a diagonalizable matrix that is not invertible (P/I)

possible

Swapping any two rows of A does not change the value of det(A) (T/F)

false

If two matrices have the same eigenvalues, then the matrices are similar (T/F)

false

If A is nxn and not invertible, then A cannot be diagonalizable (T/F)

false

If the characteristic polynomial of a 2×2 matrix has no real roots, then A must have two complex eigenvalues (T/F)

true

If the nonzero vector v is an eigenvector of nxn matrices A and B, and A ≠ B, then v is an eigenvector of A+B (T/F)

true

The steady state of a stochastic matrix is unique (T/F)

false

A 2×2 matrix A whose rank is 1 must have an eigenvalue that is equal to 0 (T/F)

true

If A is nxn and diagonalizable, then A has n distinct eigenvalues (T/F)

false

If e=0 is an eigenvalue of A, then the matrix A is non-singular (T/F)

false

If matrices A and B are similar, then A and B have the same eigenvalues (T/F)

true

Suppose A is a 3×3 matrix with two eigenvalues. If the geometric multiplicity of e1 is 1 and the geometric multiplicity of e2 is 2, then A must be diagonalizable (T/F)

true

If A is a square matrix, v and w are eigenvectors of A, then v+w is also an eigenvector of A (T/F)

false

Matrices A and B are similar and have characteristic equations that are not equal to each other (P/I)

impossible

If A is a 2×2 matrix, then det (-A) = -det(A) (T/F)

false

Any stochastic matrix with a zero entry cannot be regular (T/F)

false

A non-zero matrix A can be similar to the zero matrix (T/F)

false

If A is a real 2×2 singular matrix, then both eigenvalues of A cannot have an imaginary component (T/F)

true

A is an invertible 3×3 matrix whose eigenvalues are 0, 2, 5 (P/I)

impossible

If A is a 3×3 matrix, then det(-A) = -det(A) (T/F)

true

If q is a steady-state vector for stochastic matrix P, then the Markov Chain converges to q as k approaches infinity (T/F)

false

An eigenvalue of a matrix could be associated with two linearly independent eigenvectors (T/F)

true

Suppose A is a diagonalizable nxn matrix, and x and b are vectors in Rn. The linear system Ax=b has a solution for all b in Rn (T/F)

false

If A is a diagonalizable nxn matrix, then rank(A)=n (T/F)

false

A 2×2 matrix A with characteristic polynomial e²+1 has two real eigenvalues (T/F)

false

If matrix A is nxn and has n distinct eigenvalues, then rankA=n (T/F)

false

If A in R² has complex eigenvalues, then |e1|=|e2| (T/F)

true

If A is an nxn matrix and A-3I is non-singular, then 3 is an eigenvalue of A (T/F)

false

If an nxn matrix is invertible, then it has an linearly independent eigenvectors (T/F)

false

Suppose A is a 2×2 matrix that rotates vectors by pi/4 radians clockwise about the origin. Then A does not have real eigenvalues. (T/F)

true

Suppose v is an eigenvector of an nxn matrix A with eigenvalue 2 and an eigenvector of an nxn matrix B with eigenvalue 3. Then v is an eigenvector of A+B with eigenvalue 5 (T/F)

true

Every stochastic matrix has at least one steady-state vector (T/F)

true

If det(A)=3 and A is a 3×3 matrix, then det(-A)=3 (T/F)

false

If v is an eigenvector of nxn invertible matrix A, then v is an eigenvector of A^-1. (T/F)

true

If the stochastic matrix P is regular, the steady-state vector of the Markov chain depends on the initial state vector (T/F)

false

If v is an eigenvector for nxn matrix A with eigenvalue e=3, then y=3v is an eigenvector for A with eigenvalue e=9 (T/F)

false

If a matrix A is nxn and has n linearly independent eigenvectors, then A is diagonalizable (T/F)

true

If an eigenvalue e of an nxn matrix has algebraic multiplicity 1, then its geometric multiplicity is also 1 (T/F)

true