The homework include College Linear Algebra Questions. I will also provide the notes from the lecture.

Linear Algebra (MATH140/MA-UY3044)

Fall 2019

Homework 9: sections 6.2, 6.4, 6.5 (lectures 18, 19)

(Due Sunday 11/17/2019 before midnight)

1. (10 points) Diagonalize the following symmetric matrices into QΛQT , where Q is an

orthogonal matrix.

7 −4 4

5 0 , with eigenvalues λ = 13, 7, 1.

(a) A = −4

4

0 9

2

0

(b) B =

0

0

0

1

0

1

0

0

2

0

0

1

, with eigenvalues λ = 0, 2

0

1

0.3

0

0

2. (16 points) Let A = 0.2 0.1 0.4 .

0.1

0 0.4

(a) (4 points) Find the eigenvalues of A.

(b) (6 points) Find the eigenvectors associated with each eigenvalue of A.

(c) (5 points) Diagonalize A, and use it to compute lim Ak .

k→∞

(d) (1 points) Suppose A is an n × n matrix that is diagonalizable (so it has n linearly

independent eigenvectors). What must be true for lim Ak to exist? What is

k→∞

needed for Ak → 0? Justify your answer.

3. (3 points) Show that if A and B are similar, then det(A)=det(B).

Linear Algebra (MATH140/MA-UY3044)

Fall 2019

4. (3 points) The algebraic multiplicity of an eigenvalue λ, i.e., the number of times λ is

repeated, is always greater than or equal to its geometric multiplicity, i.e., the number

of linearly independent eigenvectors associated with λ . Find h in the matrix A below,

such that λ = 4 has a geometric multiplicity of 2, where

4

0

A=

0

0

2

2

0

0

3 3

h 3

.

4 14

0 2

5. (8 points) Label the following statements as True or False. To receive full credit,

explain why a given statement is either true or false.

(a) Two distinct eigenvectors corresponding to the same eigenvalue are always linearly

dependent.

(b) If λ is an eigenvalue of an n × n matrix A, then λ is also an eigenvalue of AT .

(c) An n×n matrix A is diagonalizable if the algebraic multiplicity of each eigenvalue

equals its geometric multiplicity.

2 1

1 0

(d) The matrices

and

are similar.

0 1

0 2

6. (10 points) Consider the sequence xk+2 = 3xk+1 − 2xk for k ≥ 0. Starting with

an initial condition x0 = 0, x1 = 1, compute x63 by finding a general formula for xk in

terms of the initial conditions.