5.2

Question 1

Use a property of determinants to show that A and A^T have the same characteristic polynomial.

Answer 1

Start with (A^T - λI) = det(A^T - λI^T) = (A - λI)^T. Then use the formula det A^T = det A

Question 2

T or F: Let A be an n x n matrix.

If λ + 5 is a factor of the characteristic polynomial of​ A, then 5 is an eigenvalue of A.

Answer 2

The statement is false. If λ + 5 is a factor of the characteristic polynomial of​ A, then -5 is an eigenvalue of A. In order for 5 to be an eigenvalue of​ A, the characteristic polynomial would need to have a factor of λ-5.

Question 3

T or F: Let A be an n x n matrix.

The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A.

Answer 3

The statement is true. It is the definition of the algebraic multiplicity of an eigenvalue of A.

Question 4

How to find eigenvectors of a 2x2?

Answer 4

Turn the eqn Ax=λx into (A-λI)x = 0. Turn A-λI into a singular matrix by subtracting the proposed eigenvalue from the main diagonal. Then get that matrix into rref and note the free variables then substitute accordingly. The eigenvector you get is in the span of the Eigenspace. NOTE: If you don’t get free variables from the rref, then the proposed eigenvalue isn’t an eigenvalue.

Question 5

How to find eigenvectors of a 3x3?

Answer 5

Turn the eqn Ax=λx into (A-λI)x = 0. Turn A-λI into a singular matrix by subtracting the proposed eigenvalue from the main diagonal. Then create zeros using row reduction. Then co factor expand to get it into a 2x2 matrix, then go.

Question 6

how to find the characteristic eqn/polynomial

Answer 6

subtract λ from the main diagonal then solve for λ. If you have repeats then that eigenvalue have a multiplicity of how many there are.