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Mathematics 221

Matrix Algebra

Solutions Manual

Department of Computer Science, Mathematics,

Physics and Statistics

University of British Columbia Okanagan

By Paul A. S. Tsopméné

April 14, 2021

About the Author.

• Title: Assistant Professor of Teaching
• Institution: University of British Columbia, Okanagan Campus
• Since: September 2020

• Introduction
• 1 Practice Problems (PB)
  • 1.1 PB1: Systems of Linear Equations
  • 1.2 PB2: Row Reduction and Echelon Forms
  • 1.3 PB3: Solutions of Linear Systems
  • 1.4 PB4: Vector and Matrix Equations
  • 1.5 PB5: Matrix Equations and Homogeneous Systems
  • 1.6 PB6: Solution Sets  Parametric Vector Form
  • 1.7 PB7: Applications of Linear Systems
  • 1.8 PB8: Linear Independence
  • 1.9 PB9: Linear Transformations
  • 1.10 PB10: The Matrix of a Linear Transformation
  • 1.11 PB11: Matrix Operations
  • 1.12 PB12: Matrix Operations (continued)
  • 1.13 PB13: The Inverse of a Matrix
  • 1.14 PB14: Properties of the Inverse of a Matrix and Elementary Matrices
  • 1.15 PB15: Elementary Matrices and Subspaces of Rn.
  • 1.16 PB16: Basis, Dimension and Rank
  • 1.17 PB17: Determinants
  • 1.18 PB18: Eigenvalues and Eigenvectors
  • 1.19 PB19: Diagonalization
• 2 Solutions to Practice Problems
  • 2.1 Solution to PB1
  • 2.2 Solution to PB2
  • 2.3 Solution to PB3
• 2.4 Solution to PB4 4 CONTENTS
• 2.5 Solution to PB5
• 2.6 Solution to PB6
• 2.7 Solution to PB7
• 2.8 Solution to PB8
• 2.9 Solution to PB9
• 2.10 Solution to PB10
• 2.11 Solution to PB11
• 2.12 Solution to PB12
• 2.13 Solution to PB13
• 2.14 Solution to PB14
• 2.15 Solution to PB15
• 2.16 Solution to PB16
• 2.17 Solution to PB17
• 2.18 Solution to PB18
• 2.19 Solution to PB19

Introduction

The present manual is primarily for students taking MATH 221Matrix Algebra at the University of British Columbia Okanagan. It contains the following.

• A wide variety of practice problems organized in such a way that students can learn gradually.
• Detailed solutions to practice problems. These are well explained step by step so that students can easily understand what is going on.
• A review of the theory. The theory related to every concept is recalled along the way, the idea being not only to show to students how to apply it but also to facilitate the understanding of solutions. This makes the manual self-contained.

Chapter 1

Practice Problems (PB)

1.1 PB1: Systems of Linear Equations

1. For each system, determine whether the given sequence is a solution to the system. Also nd the matrix of coecients and the augmented matrix.

(a) x^ −^2 y^ =^3 3 x + y = 2

(b) −^3 x^ +^4 y^ =^18 5 x − 6 y = − 27

(c)

x − y + z = 3 2 x + 3 y − z = − 5 − 3 x + 4 y + 2 z = 1

(d)

x 1 + 2 x 3 − x 4 = 7 2 x 1 + x 2 − 4 x 3 = − 9 − 5 x 2 + 10 x 4 = 15

2. Solve each system.

(a) x^ +^2 y^ =^5 2 x − y = − 10

(b)

x + y = 0 3 x − 4 y = 7

(c) 2 x^ −^3 y^ =^7 5 x + 7 y = 3

3. Solve each system.

7

8 CHAPTER 1. PRACTICE PROBLEMS (PB)

(a)

x − 2 y − z = − 6 −x + 3 y + 2 z = 11 − 2 x + 5 y + 4 z = 20

(b)

x − y + z = 2 2 x + y = − 4 − 3 x + 2 z = 14

(c)

x 1 + x 2 + x 4 = 4 x 1 + x 3 = 1 2 x 2 − x 3 + x 4 = 4 2 x 1 − x 2 + x 3 − 3 x 4 = − 5

1.2 PB2: Row Reduction and Echelon Forms

1. Determine whether each matrix is in echelon form.

A =

 , B =

C =

 , D =

 ,^ E^ =

2. Determine whether each matrix is in reduced echelon form.

A =

 , B =

C =

 , D =

 , E =

3. Write out every possible reduced echelon form of a 2 × 2 matrix. For entries that can have an arbitrary value, use an asterisk ∗ symbol.
4. Row reduce the following matrices to reduced echelon form.

A =

 , B =

C =

 ,^ D^ =

10 CHAPTER 1. PRACTICE PROBLEMS (PB)

4. Consider the following system. x + y − z = a x + 2 z = b −x − 2 y + 4 z = c

(a) Find a condition on a, b, and c such that the system is consistent. (b) When the system is consistent, nd the general solution (in terms of a, b, and c).

5. Consider the following system.

−x + 3 y + 2 z = − 8 x + z = 2 3 x + 3 y + az = b

Find conditions on a and b such that the system has no solution, one solution, or innitely many solutions.

1.4 PB4: Vector and Matrix Equations

1. Let u =

[

]

, v =

[

]

, and w =

[

]

. Compute u + v, 10 w, w − 2 u, 3 u − v − 2 w.

2. In each case, write a system of linear equations that is equivalent to the given vector equation.

(a) x 1

 (^) + x 2

(b) x 1

[

]

• x 2

[

]

• x 3

[

]

[

]

3. For each system, write the corresponding vector equation.

(a) x^1 −^ x^2 =^3 2 x 1 + 7 x 2 = − 8

(b)

2 x 1 − x 2 + x 3 = 5 − 3 x 1 + 4 x 2 + 5 x 3 = 9 8 x 1 + 6 x 3 = 11

4. In each case, determine if b is a linear combination of a 1 and a 2.

(a) a 1 =

, a 2 =

, b =

(b) a 1 =

, a 2 =

, b =

1.5. PB5: MATRIX EQUATIONS AND HOMOGENEOUS SYSTEMS 11

5. Let A =

 (^) and b =

. Determine whether b is a linear combination of the columns

of A.

6. Let a 1 =

, a 2 =

, and b =

. Determine whether b is in Span{a 1 , a 2 }.

7. Let a 1 =

, a 2 =

, and b =

α 1 2

. For what value(s) of α is b in Span{a 1 , a 2 }.

8. Let u =

[

]

, v =

[

]

. Show that Span{u, v} = R^2.

9. Consider the matrix M =

. Show that the columns of M span R^3.

10. In each case, compute the product Ax (if it is dened). If a product is not dened, explain why.

(a) A =

[

]

and x =

[

]

(b) A =

[

]

and x =

(c) A =

[

]

and x =

[

]

(d) A =

 (^) and x =

[

]

11. In each case, write the system as a matrix equation.

(a) 7 x^1 −^6 x^2 =^13 − 8 x 1 + 9 x 2 = 5

(b)

x 1 + x 2 − x 3 = 0 2 x 1 + x 3 = 1 3 x 2 − 4 x 3 = 7

1.5 PB5: Matrix Equations and Homogeneous Systems

1. Let A =

[

]

and b =

[

b 1 b 2

]

(a) Show that the system Ax = b does not have a solution for all possible b. (b) Describe the set of all b for which the system Ax = b does have a solution.

1.6. PB6: SOLUTION SETS  PARAMETRIC VECTOR FORM 13

(b)

x 1 = 5 + 6t x 2 = 10 − 8 t x 3 = t

t is arbitrary

(c)

x 1 = 1 + s + 9t x 2 = −4 + 13t x 3 = s x 4 = t

s and t are arbitrary

(d)

x 1 = 4 s − 7 t x 2 = s x 3 = −2 + 5t x 4 = − 3 x 5 = t

s and t are arbitrary

(e)

x 1 = 24 + 3s 1 − 8 s 2 + 9s 3 x 2 = s 1 x 3 = −7 + 15s 2 + 6s 3 x 4 = s 2 x 5 = s 3

s 1 , s 2 , and s 3 are arbitrary

2. Let M =

 be the augmented matrix of a system of linear equations. Write

the general solution in parametric vector form.

3. In each case, solve the system and write the general solution in parametric vector form.

(a)

x 1 − x 2 + 2 x 3 = 0 −x 1 + 4 x 2 − 7 x 3 = 0 −x 1 + 10 x 2 − 17 x 3 = 0

(b)

x 1 + 3 x 2 + 4 x 3 = 1 − 2 x 1 + x 2 + 5 x 3 = 2 7 x 1 + 7 x 2 + 2 x 3 = − 1

(c)

x 1 + 2 x 2 + x 3 + 5 x 4 = 0 2 x 1 + 4 x 2 + 3 x 3 + 7 x 4 = 0 − 3 x 1 − 6 x 2 − 5 x 3 − 9 x 4 = 0

(d)

x 1 − 2 x 2 − 3 x 3 + 4 x 4 = − 9 x 1 − x 2 − 4 x 3 + 10 x 4 = 9 − 2 x 1 + 3 x 2 + 7 x 3 − 14 x 4 = 0

4. A zero matrix is a matrix where every entry is 0. Suppose A is a 2 × 2 zero matrix. Describe the solution set of Ax = 0.

14 CHAPTER 1. PRACTICE PROBLEMS (PB)

5. A nonzero matrix is a matrix where at least one entry is nonzero. Construct a 3 × 3 nonzero matrix A

such that the vector

 (^) is a solution to Ax = 0.

1.7 PB7: Applications of Linear Systems

1. On a thin metal plate, the steady-state temperature at each point is the average of the surrounding points on the plate. Consider the plate shown in Figure 2.1. Temperatures T 1 , T 2 , and T 3 each equal

Figure 1.

the average of the temperatures of the four nearest grid points.

(a) Set up a system of equations for T 1 , T 2 , and T 3 and put it in standard form. (b) Solve the system using elementary row operations.

2. Suppose an economy has only two sectors, Goods and Services. Each year, Goods sells 80% of its output to Services and keeps the rest, while Services sells 70% of its output to Goods and retains the rest.

(a) Determine the exchange table for this economy, where the columns describe how the output of each sector is exchanged among the two sectors. (b) Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures.

3. Suppose an economy has three sectors: Agriculture, Mining, and Manufacturing. Agriculture sells 5% of its output to Mining and 30% to Manufacturing, and retains the rest. Mining sells 20% of its output to Agriculture and 70% to Manufacturing, and retains the rest. Manufacturing sells 20% of its output to Agriculture and 30% to Mining, and retains the rest. Find equilibrium prices for the annual outputs of the Agriculture, Mining, and Manufacturing sectors that make each sector's income match its expenditures.
4. The network in Figure 2.2 shows the trac ow (in vehicles per minute) over several one-way streets in downtown of a certain city during a typical early afternoon.

(a) Determine the general ow pattern for the network. (b) Determine the possible range of values of x 4. (c) When x 4 = 0, what is the minimum value of x 1?

16 CHAPTER 1. PRACTICE PROBLEMS (PB)

(a) For what value(s) of h (if any) is v 3 in Span{v 1 , v 2 }? (b) For what value(s) of h (if any) is the set {v 1 , v 2 , v 3 } linearly independent?

4. Let v 1 =

, v 2 =

, v 3 =

h

. For what value(s) of h (if any) is the set {v 1 , v 2 , v 3 }

linearly independent?

5. Let v 1 =

, v 2 =

, v 3 =

, and v 4 =

(a) Show that the set {v 1 , v 2 , v 3 , v 4 } is linearly dependent. (b) Is it possible to write v 3 as a linear combination of v 1 , v 2 , and v 4?

6. Determine by inspection if the given set is linearly dependent.

(a) v =

, S = {v}.

(b) v =

[

]

, S = {v}.

(c) v 1 =

 (^) , v 2 =

, v 3 =

 (^) , v 4 =

, S = {v 1 , v 2 , v 3 , v 4 }.

(d) v 1 =

, v 2 =

, v 3 =

, S = {v 1 , v 2 , v 3 }.

(e) v 1 =

,^ v^2 =

,^ S^ =^ {v^1 , v^2 }.

7. Let A =

[

a 1 a 2 a 3

]

be a 3 × 3 matrix and suppose that x =

 (^) is a solution to the

homogeneous equation Ax = 0.

(a) Give a linear dependence relation on the columns of A. (b) Do the columns of A span R^3? Why or why not?

8. Let A =

. Observe that a 3 = a 1 + 2a 2 , where ai is the ith column of A. Find a

nontrivial solution to the equation Ax = 0.

9. Let A be a 2 × 2 matrix with linearly dependent columns. Write out all possible reduced echelon form of A. For entries that can have an arbitrary value, use an asterisk ∗ symbol.

1.9. PB9: LINEAR TRANSFORMATIONS 17

1.9 PB9: Linear Transformations

1. Show that the transformation T : R^2 → R^3 dened by T

([

x 1 x 2

])

3 x 1 0 x 1 − x 2

 (^) is linear.

2. In each case, show that the transformation T is not linear.

(a) T : R^2 → R dened by T

([

x 1 x 2

])

[ √ 3

x 1 x 2

]

(b) T : R^2 → R^3 is dened by T

([

x 1 x 2

])

x 1 −x 1 + x 2 5 |x 1 |

(c) T : R^2 → R^2 dened by T

([

x 1 x 2

])

[

9 x 2 + 8 x 1

]

3. Suppose that T : Rn^ → Rm^ is a linear transformation, and also that w is in Span{u, v}, where u and v are vectors in Rn. Prove that T (w) is in Span{T (u), T (v)}.
4. Let u 1 =

 (^) , u′ 1 =

 (^) , u 2 =

 (^) , and u′ 2 =

. Consider the linear transforma-

tion T : R^3 → R^3 such that T (u 1 ) = u′ 1 and T (u 2 ) = u′ 2. Let v =

(a) Show that there exists a unique pair (c 1 , c 2 ) such that v = c 1 u 1 + c 2 u 2. (b) Find T (v).

5. Let e 1 =

[

]

, e 2 =

[

]

, v 1 =

[

]

, and v 2 =

[

]

. Let T : R^2 → R^2 be a linear transformation that maps e 1 into v 1 and e 2 into v 2.

(a) Find T

([

])

(b) Find T (x) for an arbitrary x =

[

x 1 x 2

]

6. Find an example of a linear transformation T and two independent vectors u, v such that T (u) and T (v) are not independent.
7. Let u, v be vectors in Rn^ such that the set {u, v} is linearly independent. Let T : Rn^ → Rm^ be a linear transformation such that the set {T (u), T (v)} is linearly dependent. Show that the equation T (x) = 0 has a nontrivial solution.
8. Let A be a matrix of size 7 × 11 , and let T : Ra^ → Rb^ be the linear transformation dened by T (x) = Ax. What is the value of a? What is the value of b?
9. Let A =

, u =

[

]

, and b =

. Let T : R^2 → R^3 be the transformation dened

by T (x) = Ax.

1.11. PB11: MATRIX OPERATIONS 19

1.11 PB11: Matrix Operations

1. Let

A =

[

]

, B =

[

]

, C =

 (^) , and D =

Compute each of the following expressions if they are dened. If an expression is undened, explain why.

(a) A + B (b) C + D (c) A + C (d) C + D + B (e) 2 C (f) 3 C − 2 D

2. Let A =

[

]

and B =

. Compute A − 2 I 2 and 7 I 3 − B.

3. Consider the matrices A and B from Question 1. Compute − 5 A + 2B − 3 I 2.
4. Solve the the equation X +

[

]

[

]

, where X is a 2 × 2 matrix.

5. Let

A =

[

]

, B =

[

]

, C =

 , D =

E =

[

]

, and F =

Compute each of the following expressions if they are dened. If an expression is undened, explain why.

(a) AB (b) BA (c) BC (d) CB (e) CD (f) DC (g) CE (h) EC (i) DF

20 CHAPTER 1. PRACTICE PROBLEMS (PB)

6. Let A and B be matrices such that A is of size 5 × 7 and the product AB is of size 5 × 11. What is the size of B?
7. Let A and B be matrices such that the product BA is of size 17 × 32.

(a) How many rows does B have? (b) How many columns does A have?

8. Let A be an m × n matrix, and let B be an n × p matrix. Show that every column of the product AB is a linear combination of the columns of A.
9. Let f : Rp^ → Rn^ and g : Rn^ → Rm^ be two transformations. The composition of f and g, denoted g ◦ f , is the transformation g ◦ f : Rp^ → Rm^ dened as (g ◦ f )(x) = g(f (x)) for every x in Rp. Suppose f and g are both linear.

(a) Show that g ◦ f is a linear transformation. (b) Let A be the standard matrix of f , and let B be the standard matrix of g. Express the standard matrix of g ◦ f in terms of A and B.

1.12 PB12: Matrix Operations (continued)

1. Let A, B, and C be n × n matrices. Assume that A and B commute. Also assume that A and C commute. Show that A commutes with BC.
2. Let A =

[

]

and B =

. Compute A^2 and B^3.

3. If A =

[

]

, B =

, compute A^2 , AB, BA, and B^2 when they are dened.

4. Let

A =

[

]

, B =

 , C =

[

]

, D =

Find AT^ , BT^ , CT^ , and DT^.

5. Let A =

[

]

. Compute AAT^ − 5 I 2 and 8 I 3 − AT^ A.

6. If A and B are matrices, can you conclude in general that (A + B)^2 = A^2 + 2AB + B^2 ?. If this is not true in general, give a specic example of A and B that makes the equality false.
7. A student tries to solve the matrix equation X^2 + BX = XC (where all of these are n × n matrices) as follows: X^2 =XC − BX X^2 =X(C − B) X^2 − X(C − B) = X(X − (C − B)) = X = 0 or X = C − B