If A is a square matrix, which of the following statements is true about its eigenvalues?

• • They can be complex numbers

• • They are always real numbers

• They are always positive

• • They must all be distinct.

• n-1

• n

• 2n-1

• Ifinity

If λ is an eigenvalue of the matrix A, which of the following statements is true?

• λ is the sum of the eigenvalues of A.

• The eigenvectors corresponding to λ are always orthogonal.

• λ is a solution to the characteristic polynomial of A.

• λ must be an integer

If a matrix A is diagonalizable, which of the following is necessarily true?

• • All eigenvalues are distinct

• • The matrix is symmetric

• • It has a complete set of linearly independent eigenvectors

• It has at least one positive eigenvalue

The sum of the eigenvalues of a matrix A is equal to:

• • The trace of A

• • The determinant of A

• • The product of the eigenvalues.

• • The inverse of the matrix

Which of the following matrices does NOT have real eigenvalues?

• • A symmetric matrix.

• • A skew-symmetric matrix

• • A Hermitian matrix

• • A real diagonal matrix

If A is a 2×2 matrix with eigenvalues λ1 and λ2​, what is the product of the eigenvalues equal to?

• • The sum of the eigenvalues

• • • The determinant of A

• • The trace of A

• The inverse of the matrix

If a matrix A has an eigenvalue λ=0, what can be said about the matrix?

• • A is invertible

• • The determinant of A is zero

• All eigenvalues are zero

• • The matrix is symmetric

For a matrix AAA, if λ\lambdaλ is an eigenvalue with a multiplicity kkk, which of the following statements is true?

• The algebraic multiplicity of λ is k

• • The geometric multiplicity of λ is k

• • A must have k distinct eigenvectors corresponding to λ

• None of the above

Which of the following properties holds for eigenvectors corresponding to distinct eigenvalues of a matrix A?

• • They are always linearly dependent

• • They are always orthogonal if A is symmetric

• • They cannot form a basis

• They are unique