Matrix Quiz Week 3 I - CSC C

If λ is an eigenvalue of a matrix A, what is the relationship between the eigenvalues of A and AT (the transpose of A)?

• They are equal.

• They are all negative.

• They are all positive.

• They are complex conjugates.

If A is a n×n matrix with all eigenvalues having a magnitude less than 1, what can be concluded about Ak as k→∞?

• Ak diverges to infinity.

• Ak converges to the zero matrix.

• Ak approaches the identity matrix.

• Ak oscillates indefinitely.

What is the characteristic polynomial of a n×n matrix A?

• det (A−λI)

• det (A+λI)

• λn−trace(A)

• The sum of the eigenvalues.

If the eigenvalues of a matrix AAA are all real and positive, what can be said about AAA?

• A is positive definite.

• A is singular.

• A must be symmetric.

• None of the above.

What can be inferred about the eigenvectors of a matrix with complex eigenvalues?

• The eigenvectors must be complex.

• The eigenvectors are always real.

• The eigenvectors are orthogonal.

• There are no eigenvectors.

Which of the following transformations preserves the eigenvalues of a matrix A?

• Transposing A.

• Taking the inverse of A.

• Adding a scalar multiple of the identity matrix to A.

• Both A and C.

Which of the following statements is true regarding the eigenvalues of the product of two matrices A and B (both n×n)?

• The eigenvalues of AB are the product of the eigenvalues of A and B.

• The eigenvalues of AB are equal to the eigenvalues of BA.

• The eigenvalues of AB are always real.

• The eigenvalues of AB are equal to the sum of the eigenvalues of A and B.

What can be concluded about the eigenvalues of a matrix that is both symmetric and positive definite?

• All eigenvalues are real and positive.

• All eigenvalues are complex.

• At least one eigenvalue is negative.

• Eigenvalues can be zero.

If a matrix A has an eigenvalue of zero, what can be inferred about the matrix A?

• A is invertible.

• A is singular.

• A must be diagonal.

• A has full rank.

Which of the following statements is true regarding the eigenvalues of a diagonal matrix?

• The eigenvalues are the entries on the diagonal.

• The eigenvalues are always zero.

• The eigenvalues are the sum of the diagonal entries.

• The eigenvalues are the product of the diagonal entries.