S10

Question 1

The sum of two eigenvectors of a matrix A is also an eigenvector of A.

Answer 1

False: (1,0) and (0,1) are both eigenvectors of A, but (1, 1) is not.

(It is true if the two eigenvectors belong to the same eigenvalue, though.)

Question 2

A is singular if and only if 0 is an eigenvalue of A.

Answer 2

True

0 is an eigenvalue if and only if 0 = det(A−0·I) = det(A) if and only if A is singular.

Question 3

If v₁ and v₂ are linearly independent eigenvectors of A, then they correspond to distinct eigenvalues.

Answer 3

False

We have seen two-dimensional eigenspaces, which contain two linearly independent eigenvectors that correspond to the same eigenvalue.

Question 4

If A is invertible, then it is diagonalizable.

Answer 4

False

A is invertible, but not diagonalizable.

Question 5

If A is singular, then it is not diagonalizable.

Answer 5

False

A is singular, and diagonalizable because it is already diagonal.

Question 6

If A has fewer than n distinct eigenvalues, then A is not diagonalizable.

Answer 6

False

Identity Martix (Shortest answer ever.)

Question 7

Each eigenvector of an invertible matrix A is also an eigenvector of A⁻¹.

Answer 7

True

If Av = λv, then (^₁/_λ) v = vA⁻¹, so v is an eigenvector of A⁻¹ with eigenvalue (¹/_λ).