5.1

Question 1

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

The eigenvalues of a matrix are on its main diagonal.

Answer 1

The statement is false. If the matrix is a triangular​ matrix, the values on the main diagonal are eigenvalues.​ Otherwise, the main diagonal may or may not contain eigenvalues.

Question 2

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

Finding an eigenvector of A may be​ difficult, but checking whether a given vector u is in fact an eigenvector is easy.

Answer 2

The statement is true. Checking whether a given vector u is in fact an eigenvector is easy because it only requires checking that Bold u is a nonzero vector and finding if Au is a scalar multiple of u.

Question 3

what’s an eigenvector?

Answer 3

It’s the product of Au when it’s a scalar multiple of u.

Question 4

what’s an eigenvalue?

Answer 4

it’s the scalar of u where Au is a scalar multiple of u.

Question 5

how do we show that a # is an eigenvalue of a matrix?

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 get that matrix into rref and note the free variables then substitute accordingly. If you get some eigenvector, it’s an eigenvalue NOTE: If you don’t get free variables from the rref, then the proposed eigenvalue isn’t an eigenvalue.

Question 6

T or F: the eigenvalues of a triangular matrix are the entries on its main diagonal.

Answer 6

True

Question 7

T or F: 0 can be only an eigenvector and eigenvalue?

Answer 7

False. 0 can only be an eigenvalue not an eigenvector.