Mathematics 244 – Linear Algebra Additional Review True-False Questions – Exam 3 April 23, 2004

For the true statements, give a short proof. For the false statements, give a reason or a counterexample.

A) If A is a 3 × 3 matrix whose characteristic polynomial has a double root, then A is not diagonalizable.

B) If T : R^{2} → R^{2} and there is a non-zero vector x that satisfies T^{2}(x) = 4x, then T (x) = ±2x.

C) Every 3 × 3 matrix has at least one eigenvalue λ ∈ R.

D) If λ is an eigenvalue of (A − sI)^{−1}, where s ∈ R, then s + \frac{1}{λ} is an eigenvalue of A.

E) The eigenvalues of ( \begin{matrix}a & b \\ b & a \end{matrix} ) are λ = a + b and λ = a − b.

F) If the linear mapping T : R^{3} → R^{3} has eigenvalues λ = 1, 2, and x_{1}, x_{2} are linearly independent eigenvectors for λ = 1, while y is an eigenvector for λ = 2, then E = {x_{1}, x_{2}, y} is linearly independent.

G) If A is an n × n matrix with linearly independent rows, then det(A) = 0.

H) Let V be a vector space of dimension n and T : V → V be linear. If [T]_{α}^{α} = I_{n} (the n × n identity matrix) with respect to some basis α for V, then T is the identity mapping.