Mathematics IA Algebra Practice Questions
Questions marked with a * are more challenging, or contain further applications which are not examinable.
1. Matrices and Linear Equations
Matrices


1
1
8 4 2
5 −4
0 , B =
1. Let A =  2
, C =
, D = 3 5 7
2 4 6
0 0
−1 −4
−2
and E =
.
−1
Evaluate each of the following matrices if they exist:
(i) At (ii) AB (iii) BAC (iv) A + C (v) A + B t (vi) ED
(vii) AEDB t C t (viii) C 3
a b c
2. Let A =
. Find expressions for AAt and At A.
d e f
3. Let A, B, C and D be matrices which satisfy the following conditions:
AB t C 2 Dt is defined and is a 3 × 3 matrix, A has 7 columns, and B
has 5 rows. Find the sizes of each of the matrices A, B, C and D.
4. Let A be an m × n matrix, and suppose there exist n × m matrices C
and D such that CA = In and AD = Im . Prove that C = D.
5. A square matrix A is said to be symmetric if A = At . Let A be an
m × n matrix. Prove that AAt and At A are symmetric matrices.
3 8
1
2
. Find all possible matrices C
and B =
6. Let A =
1 −2
−1 −2
such that both AB = AC and BA = CA.
7.* Two square matrices A and B are said to commute if AB = BA.
Find all 2 × 2 matrices A with commute with every 2 × 2 matrix.
a b
; choose some very simple matrices B — for
(Hint: let A =
c d
instance matrices with almost every entry equal to 0 — and investigate
what conditions must be satisfied by a, b, c and d in order to have
AB = BA.)
Linear Equations
8. Solve the following systems of linear equations:
(a)
2x1 + 6x2
4x1 − 3x2
= 12
= 19
(b)
1
x1 − 3x2 + 2x3
2x2 − x3
x1 + x3
= -10
=6
=0
9. Solve the following systems of linear equations:
(a)
2x1 + 6x2
x1 + 3x2
=3
=1
(b)
x1 + 2x2
x2 − x3
=4
=2
10. Consider the system of linear equations,
x1 − 2x2 = 1
3x1 + ax2 = 4
(a) If this system has a unique solution, (x1 , x2 ) = (5, 2), then what
is a?
(b) If this system has no solutions then what is a?
Gauss-Jordan Elimination
11. For each of the following matrices in reduced row echelon form, give
the set of solutions to the corresponding system of equations (you can
call the variables x1 , x2 , . . . ).




1 0 | 0
1 0 0 | 0
(a) 0 1 | 1 (b) 0 1 0 | 1
0 0 | 0
0 0 0 | 0


1 2 0 3 0 | 4
(c) 0 0 1 2 0 | 3 (d) 1 4 8 3 | −1
0 0 0 0 1 | 2
12. For each matrix below, use
reduced row echelon form:

0 1 |
(a) 0 0 |
1 0 |
row operations to put the matrix into



1 0 1 | 1
0
1 (b) 0 2 0 | 2
1 1 0 | 1
0
1 2 3 | 4
for some α 6= 0
(c)
α 0 1 | 2
13. Find the solution space of the following systems of linear equations:
(a) 4x + 2y + 5z = 8
(b) 4x + 2y + 4z = 8
2x + 4y + 4z = 1
x + 4y + 4z
−2y − z = 2
=1
2x − 6y − 3z = 6
14. If a 5 × 5 augmented matrix for a system of 5 linear equations in 4
unknowns is row reduced:
(a) How many pivots must there be for a unique solution?
(b) How many zero rows must there be for a unique solution?
2
15. For which values of c
have
(i) no solutions, (ii) a
When solutions exist,
c).
(a) x + cy
cy
does each of the following system of equations
unique solution, (iii) infinitely many solutions?
write down what they are (possibly in terms of
=0
(b)
=1
x + cz = c
x + cy + cz = 0
x + cz + z = 1
16. Suppose 3 foods are used to make a meal. The foods have the following
units per gram of vitamin B and vitamin C respectively.
Vitamin B Vitamin C
Food 1
12
18
Food 2
8
10
Food 3
4
2
By setting up a system of linear equations, determine if it is possible
to combine these foods into a meal of 500 grams with precisely 2000
units of Vitamin B and 1000 units of Vitamin C. If this is possible,
how many ways can it be done?
17.* Let A be an m × n matrix. If the system of equations Ax = b has
a solution for every b, show that there is an n × m matrix D so that
AD = Im .
18.* Let A be an m × n matrix and b be a vector in Rm . Prove that there
cannot be exactly two solutions to Ax = b.
Linear Combinations
19. Decide whether each of the following vectors is in the span of
{(1, 0, 1, 2, 0), (2, 1, 1, 0, 1), (0, 2, 2, 1, 1)}
and where possible write the vector as a linear combination of the
vectors in the span:
(a) (7, −2, 1, 4, 0) (b) (4, 3, 5, 5, 1) (c) (−1, −2, −1/2, 5/2, −3/2)
(d) (−1/2, 13, 17/2, −7/2, 15/2).


1 5 2
20. Let A = −1 9 5 , and let W be the set of all linear combinations
1 3 1
of the columns of A.
(a) Show that (4, 10, 2)t is in W (you should be able to do this by
inspection, that is, without any row operations).
(b) Solve the homogeneous system Ax = 0.
(c) Given that (1, 2, −1)t satisfies Ax = b, what is the vector b?
(d) Write down the general solution of Ax = b, without directly
solving this system.
3

 
1
3 −4
b1



2 −6 and b = b2 .
21. Let A = 3
−5 −1 8
b3

(a) Show that the equation Ax = b is not consistent for all b ∈ R3 .
(b) Find an equation for the set of all b for which Ax = b is consistent. Describe the set geometrically.
(c)* Use your answer to part (b) to deduce that
     
   
3
−4 
0 
 1
 1









3 , 2 , −6
0 , 1 .
span
= span




−5
−1
8
1
−2
22.* Let u, v and w be vectors in Rn . Prove that w ∈ span{u, u − v, v −
w}. (In other words, you need to prove that w is a linear combination
of u − v, v − w and u.)
23.* Let v1 , v2 , . . . , vk be a sequence of vectors in Rn . Suppose that
vk+1 ∈ span{v1 , . . . , vk }. Prove that
span{v1 , . . . , vk } = span{v1 , . . . , vk , vk+1 }.
(In other words you need to show that every linear combination of the
vectors v1 , . . . , vk is also a linear combination of the vectors v1 , . . . ,
vk , vk+1 and vice versa).
Homogeneous Equations
24. (a) Use Gauss-Jordan elimination to solve the following homogeneous
system
x1 − x2 − 4x3 = 0,
x1 + x2 + 4x3 = 0,
2x2 + x3 = 0.
(b) Find by inspection a particular solution of the system
x1 − x2 − 4x3 = −4,
x1 + x2 + 4x3 =
6,
2x2 + x3 =
3,
and hence write down the general solution, using the results of
(b).
Elementary Matrices and the Inverse Matrix
25. Let A,B and C be invertible n × n matrices. Find expressions in terms
of A, B, C and their inverses for the inverses of the following matrices:
(i) ABC (ii) AB −1 A (iii) 3At BC 2 (iv) −BA−1 C t A
26. Suppose that A and B are invertible matrices. Is (A + B)−1 = A−1 +
B −1 ? Give reasons to justify your answer.
4
27. Which of the following are elementary matrices?
1 0 0
0 1
1 100
0 −1
1 0
(i)
(ii)
(iii)
(iv)
(v)
0 −1 0
1 0
0 1
1 0
0 0

0
28. Row reduce A = 2
1
down the elementary

1 0
2 2 to reduced row echelon form and write
0 2
matrix corresponding to each row operation.
29. Give the inverse of each of the following elementary matrices:






1 0 0
0 0 1
1 0 0
(i) 0 34 0 (ii) 0 1 0 (iii) 0 1 0
0 0 1
1 0 0
0 −6 1
30. Determine E a product of elementary matrices which when premultiplying A performs Gauss-Jordan pivoting on the (3, 3) entry of


1 0 −3 5
A = 0 1 3 4 .
0 0 2 6
Check E by computing EA.
31. For each of the following matrices find the inverse by using elementary
row operations. Write down the elementary matrix corresponding to
each row operation that you use, and hence in each case express both
the matrix and the inverse as products
matrices.
 of elementary 
0 0 1
1 0 0
1 3
(c) 1 1 1.
(a)
(b) 0 0 1
4 3
0 1 2
1 2 3
32. Find all values of α for which the following matrix is not invertible.


1
4 −3
A = −2 −7 6  .
−1 3
α
33. Let A and B be matrices of the same size and suppose that B is
invertible.
(a) Prove that A is invertible if and only if AB is invertible.
(b)* Hence prove that either A + B and I + AB −1 are both invertible
or else both not invertible.
34. Let a,b, c be fixed non-zero constants.


a 0 0
(a) Find the inverse of 0 b 0 in terms of a, b, c.
0 0 c


a a 0
(b) Find the inverse of 0 a a.
a 0 a
5
35. True or False? Examine each of the following statements carefully and
decide whether they are true or false. Give a short reason for your
decision in each case.
(a) Let A and B be square matrices such that AB = O. If A is
invertible then B = O.
(b) Let A and B be invertible matrices of the same size. Then (A +
B)−1 = A−1 + B −1 .
(c) If A is an m × n matrix, then there is an invertible matrix E such
that EA is in reduced row echelon form.
(d) The matrix


1 −1 2
3
7 0


0
1 5
0 1 −2


2 0
4
−3 1 8 


A=
3
7 0
1 −1 2



4
8
11
−21
0
−7


3 5 −6
2
1 4
is invertible.
36. Prove that if A and B are row equivalent matrices (i.e. you can get
from one to the other by a series of elementary row operations) then
the equations Ax = 0 and Bx = 0 have the same solutions. (Hint:
the proof should use elementary matrices)
37. Let A be an n × n matrix with two identical columns. Explain why A
is not invertible.
38. Suppose that A is an invertible n×n matrix satisfying A3 −3A+2I = 0.
Find an expression for A−1 in terms of A and I.
2. Determinants
Basic Definitions
39. Calculate the following determinants (answers should be formulas in
terms of a and b):
1 2 a 2
0
a
b
a 1 1
1 0 a a
a
0
.
(c)
(b) b
(a) 1 a 0
0 0 0 1
a 1 0
−a −b −a
1 a b b
a 5 3 b
0 6 0 0
a b
(your answer should be
40. Given that
= 4, evaluate
0 7 −1 0
c d
c 8 4 d
a number).
6
41. Let Jn be the n × n matrix all of whose entries
example,

1
1 1
J1 = [1], J2 =
, J3 = 1
1 1
1
are equal to 1. For

1 1
1 1 .
1 1
Prove that if n > 1, then the matrix In − Jn is invertible with inverse
(In − Jn )−1 = In −
1
Jn
n−1
(here In is the n × n identity matrix).
42. Show that all of the determinants D2 , D3 , D4 , . . . Dn , . . . are equal to
1:
1 1 1 1
1 1 1
1 2 2 2
1 1
...
D3 = 1 2 2
D4 =
D2 =
1 2 3 3
1 2
1 2 3
1 2 3 4
1 1 1 1 ... 1
1 2 2 2 ... 2
1 2 3 3 ... 3
Dn =
1 2 3 4 ... 4
. . . . ... .
1 2 3 4 ... n
Row Operations On Determinants
a b c
43. Given that d e f = 5, find
g h i
d e f
(i) g h i
a b c
a − 4g
(iii) b − 4h
c − 4i
 2
a

44. Let A = b2
c2
(ii)
d
e
f
2a + d 2b + e 2c + f
g
h
i
d g
e h .
f i

a 1
b 1.
c 1
(a) Use row operations to show that
det A = (b − a)(c − a)(b − c).
(b) Calculate the adjoint and inverse of A when a, b, c are distinct.
(c) Use (b) to find α, β, γ so that y = f (x) = αx2 + βx + γ has
f (1) = 1, f (2) = 0, f (3) = 2.
7
45.* The numbers 22253, 62951, 54043, 70499 and 19635 are all divisible
by 17. Use this fact to show that if
2
6
D= 5
7
1
2
2
4
0
9
2
9
0
4
6
5
5
4
9
3
3
1
3
9
5
then the number D is divisible by 17, without evaluating the determinant.
General Properties of Determinants
46. Suppose A and B are n × n matrices with det A = 4 and det B = −3.
Find each of the determinants below:
(a) det(AB)
(b) det(A2 )
(c) det(B −1 A)
(d) det(2A)
(e) det(3B t )
(f) det(AAt )
47. What can you say about det(A) if a square matric A satisfies
(a) A2 = A (such a matrix is called idempotent).
(b) Am = 0 for some m > 1 (such a matrix is called nilpotent).
48.*
(a) Let A be a 2 × 3 matrix. Then AT A is a 3 × 3 matrix. Show that
det(AT A) = 0.
(b) Suppose again that A is a 2 × 3 matrix; is it necessarily true that
det(AAT ) = 0?
The Adjoint


1 2 3
49. Consider the matrix A = 4 5 6 .
7 8 10
(a) Find the matrix [Aij ] of minors of A.
(b) Find the matrix [Cij ] of cofactors of A.
(c) Find the adjoint, adj(A), of A.
(d) Find det(A).
(e) Hence write down the inverse of A.
50.* Let A = [aij ] be an n × n matrix. If det(A) = 1 and each entry of A
is an integer (i.e. aij ∈ Z for all i, j), prove that all of the entries A−1
are also integers.
8
3. Optimisation and Convex Sets
Convex Sets
51. (a) Write down the definition of a convex set.
(b) Define a vertex of a convex set.
52. Sketch the following sets in R2 and determine the vertices.
(a)
(b)
(c)
{(x1 , x2 ) | x1 + 3x2 ≤ 6, 2x1 − x2 ≥ 4, x1 ≥ 0, x2 ≥ 0}
{(x1 , x2 ) | x1 + x2 ≥ 2, x1 − x2 ≤ 4, x1 ≥ 1, x2 ≥ 0}
{(x1 , x2 ) | −x1 + 4x2 ≤ 5, x1 − 4x2 ≤ 2, x1 + 2x2 ≥ 2}
53. Sketch the following sets in R2 and hence determine whether they are
bounded or unbounded, and whether they are convex or not. In the
case where the set is not convex, give an example of a line segment
where the definition of convexity breaks down.
(a)
(b)
(c)
(d)
{(x, y) | x + y ≥ 1, x − y ≤ 2, x ≥ 0}
{(x, y) | 1 ≤ |x| ≤ 2, |y − 3| ≤ 2}
{(x1 , x2 ) | x21 + x22 ≤ 4 and x1 + x2 ≤ 1}
{(x1 , x2 ) | x21 + x22 ≥ 1 and x1 + x2 ≤ 4}
54. Using the definition of convexity, show that the set
C = {(x1 , x2 ) ∈ R2 | (5, −1) · (x1 , x2 ) ≤ 7}
is convex.
55. Let u and v be the points (−2, 1, 0) and (1, −5, 6) in R3 .
(a) Let x be the point (−1, −1, 2). Find t such that
x = (1 − t)u + tv.
Is x on the line segment joining u and v?
(b) Is the point y = (−4, 5, −4) on the line segment joining u and v?
Is y on the line through u and v?
(c) Is the point z = (2, −4, 4) on the line segment joining u and v?
Is z on the line through u and v?
56.* Let C = {(x1 , x2 ) ∈ R2 | x1 + x2 ≤ 1, x1 ≥ 0, x2 ≥ 0}. Consider
v = (1, 0). Suppose there exist u = (u1 , u2 ) and w = (w1 , w2 ) in C,
and t ∈ (0, 1), such that v = (1 − t)u + tw.
(a) Show that if u1 < 1 or w1 < 1, then (1 − t)u1 + tw1 < 1. This
means u1 = w1 = 1. Why?
(b) Hence or otherwise show that v is a vertex.
57.* Let S = {u1 , . . . , un } be a set of vectors in Rn . A vector w is said
to be a convex combination of the vectors u1 , . . . , un if w is a linear
combination w = x1 u1 + · · · + xn un where the scalars x1 , . . . , xn are
non-negative real numbers and x1 + · · · + xn = 1. Let Conv(S) be the
set of all convex combinations of the vectors in S. Use the definition
of convexity to show that Conv(S) is a convex set.
9
Methods for Solving Optimisation Problems
58. We wish to find the minimum value of g(x, y) = 5x/2 + 5y subject to
the constraints
3x + 2y ≥ 36
3x + 5y ≥ 45
x≥0
y≥0
Sketch the feasible region, identify its vertices, and find the vertex or
vertices at which the smallest value of g(x, y) is taken.
59. Consider the convex set given by
2x1 + x2 ≤ 8
2x1 + 3x2 ≥ 12
x1 ≥ 0
x2 ≥ 0
(a) Add slack variables to write this in standard form
C = {x ∈ Rn | Ax = b, x ≥ 0}.
(b) Find the basic solutions of Ax = b and indicate those which are
feasible.
(c) Check your answer to the previous point by sketching the feasible
region.
(d) Find the maximum value of f (x) = x1 − x2 over C and write
down the corresponding optimal choice for x.
60. Consider the region given by
x≤4
x+y ≥1
x−y−z ≤8
x, y, z ≥ 0
(a) Formulate this problem using slack variables s1 , s2 and s3 . Write
down the basic solution obtained by choosing the slack variables
as pivots and state whether it is feasible.
(b) Find the basic solution corresponding to pivots z, s1 and s2 and
state whether is is feasible.
(c) Find the basic solution corresponding to pivots y,s1 and s3 and
state whether it is feasible.
10
Formulation of Optimisation Problems
61. A pet food manufacturer produces two types of food: regular and
premium. A 20kg bag of regular food requires 2 hours to prepare
and 3 hours to cook; a 20kg bag of premium food requires 2 hours to
prepare and 5 hours to cook. The materials used to prepare the food
are available 8 hours per day and the oven used to cook the food is
available 15 hours per day. The profit on a 20kg bag of regular food
is $42 and on a 20kg bag of premium food it is $50. Determine how
many bags of each type of food should be made to maximise the profit
(fractional bags can be produced – they just sell them in smaller bags).
62. A trucking company owns two types of trucks. Type A has 20 cubic
metres of refrigerated space and 30 cubic metres of non-refrigerated
space. Type B has 20 cubic metres of refrigerated space and 10 cubic
metres of non- refrigerated space. A customer wants to haul some
produce a certain distance and will require 160 cubic metres of refrigerated space and 120 cubic metres of non-refrigerated space. The
trucking company figures that it will take 300 litres of fuel for the type
A truck to make the trip and 200 litres of fuel for the type B truck.
Find the number of trucks of each type that the company should allow
for the job in order to minimise fuel consumption
(a) graphically, and
(b) algebraically (i.e. by using slack variables).
63. Two oil refineries produce three grades of gasoline, A, B and C. At
each refinery the various grades of gasoline are produced in a single
operation so that they are in fixed proportions. Assume that one
operation at Refinery 1 produces 1 units of A, 1 unit of B, and 3 units
of C. One operation at Refinery 2 produces 1 unit of A, 4 unit of
B, and 2 units of C. Refinery 1 charges $400 for one operation, and
Refinery 2 charges $300 for one operation. A consumer needs 170 units
of A, 200 units of B, and 360 units of C. How should the orders be
placed if the consumer’s needs are to be met most economically? (Use
graphical means to solve).
64. Formulate as an optimisation problem, but do not solve:
A manufacturer of a particular product is planning the production
schedule for the months of October, November, and December. The
demands for the product are for 70 units in October, 90 units in
November, and 120 units in December. The industrial plant has the
capacity to produce at most 100 units per month. It costs $150 to
produce each unit in October, and $160 in each of November and December. As well, each unit held over in storage from one month to the
next, attracts a storage fee (inventory cost). There is no storage cost
in the month of production, but there is a cost of $30 in any following
month. (Thus the storage cost for one unit is found by multiplying
the number of items still in storage on the first day of the month by
$30.) The manufacturer plans to have no items in storage at the beginning of October, and none in storage at the end of December. The
11
manufacturer wishes to determine the number of units which should
be produced in each month in order to minimise costs. (Fractional
units can be produced.)
4. The Vector Space Rn
Linear Dependence and Independence
65. Let u = (2, 1, 4, 3), v = (3, 4, 1, 2), w = (1, 2, −1, 0).
(a) Are the vectors {u, v, w} linearly independent? Justify your
answer.
(b) Are the vectors {v, w} linearly independent? Justify your answer.
(c) Are the vectors {v, w, 0} linearly independent? Justify your
answer.
(d) Are the vectors {u, w, 5u − 3w} linearly independent? Justify
your answer.
66. Which of the following lists of vectors are linearly independent:
(a) {(3, 4), (4, 3)}
(b) {(2, 1, −3, 6), (5, 3, 7, 8), (1, 1, 13, −4)}
(c) {(2, 1, 3), (2, −2, −5), (7, 3, 9)}
(d) {e1 , e1 + 2e2 , e1 + 2e2 + 3e3 , . . . , e1 + 2e2 + 3e3 + · · · + nen }
where ei ∈ Rn is the vector which has 1 in the i-th place and 0
everywhere else.
67. For what value(s) of d are the following sets of vectors linearly dependent? Justify your answers.
(a) {(1, −1, 4)), (3, −5, 5), (−1, 5, d)}
(b) {(3, 7, −2), (−6, d, 4), (9, 1, −4)}
68. Determine, with reasons, if the following statements are (A) always
true, (B) always false, or (C) might be true or false.
(a) If v1 , ..., v5 ∈ R5 and v1 = v2 + v3 then the set {v1 , ..., v5 } is
linearly dependent.
(b) If v1 and v2 ∈ R2 lie on the same straight line through the origin,
then the set {v1 , v2 } is linearly dependent.
(c) If {v1 , v2 , v3 } is linearly dependent, then so is {v1 , v2 }.
(d) If {v1 , v2 } is linearly independent, then so is {v1 + v2 , v1 − v2 }.
(e) There is a set {v1 , . . . , v5 } of linearly independent vectors in R4 .
69. What is the definition of a subspace of Rn ?
12
Subspaces
70. Which of the following are subspaces of Rn for some n? Give a reason
for each answer.
(a) W1 = {(x, y, z) ∈ R3 | 2x − y = 3z + x = 0}
(b) W2 = {(x, y, z, w) ∈ R4 | x + y = zw}
(c) W3 = {(α, β, γ) ∈ R3 | γ = 0}
(d) W4 = {(x, y, z) ∈ R3 | x + y + z = 1}
(e) W5 = {(x, y) ∈ R2 | x2 + y 2 = 1} ∪ {(0, 0)}
(f) W6 = {(x, y) ∈ R2 | x2 + y 2 = 0}.
71. Let A be an m × n matrix. Show that
(a) R = {y ∈ Rm | Ax = y for some x ∈ Rn } is a subspace of Rm
(b) K = {x ∈ Rn | Ax = 0} is a subspace of Rn .
72. Suppose that U, V ⊂ Rn are subspaces of Rn . Prove that the intersection U ∩ V is a subspace of Rn .
73. Consider the following two subsets of R4 :
U = {(x, x + 1, y + z, 2x)| x, y, z ∈ R}
V = {(x, x + y + 1, z, 2x)| x, y, z ∈ R}
Although these sets look similar, one is a subspace and one is not.
(a) Explain why U is not a subspace of R4 .
(b) Show that V = span{v1 , v2 , v3 }, where v1 = (1, 1, 0, 2), v2 =
(0, 1, 0, 0) and v3 = (0, 0, 1, 0). Deduce that V is a subspace of
R4 .
Basis
0
a
is
74. What constraints must be imposed on a, b ∈ R, so that
,
b
0
a basis for R2 .

 
x





 
y


75. Find a basis for W =   x − 2y + w = 0, 3x − z + 3w = 0 .
z






w
76. Find a basis for the subspace U = {x ∈ R4 | Ax = 0} where


1 1 1 1
A = 2 0 2 0 .
3 1 3 1
77. Show that if {u, v} is a basis for a subspace W , then so is {u + v, v}.
13
78.* Let v1 , . . . , vr be a sequence of vectors in a subspace W of Rn with
the property that every vector in W can be written uniquely as a linear
combination of the vectors v1 , . . . , vr . Prove that the vectors v1 , . . . ,
vr form a basis for W .
79. Explain why the following are not bases for the indicated subspaces.
6
−6
x
(a)
,
for V =
x + 6y = 0 ;
−1
1
y
 
 

 4 
 x

(b) 1 for W = y  x − 2y − 2z = 0 ;




1
z
   
2 
 4
(c) 1 ,  0  for W as in part (b).


1
−1
     
4
5 
 1





4 , 2 , 6 for R3 .
(d)


2
1
3
5. Eigenvalues and Eigenvetors
Eigenvalue Problem
80. Let A be an n × n matrix.
(a) Define what it means for x ∈ Rn to be an eigenvector of A with
eigenvalue λ.
(b) Define the characteristic polynomial of A.
81. Find all eigenvalues and eigenvectors for the following matrices.
0
3
(a) A1 =
6 −3


2 1 0
(b) A2 =  1 3 1 
0 1 2


1 5 5
(c) A3 =  0 3 2 
0 2 3
82. Show that if A has eigenvalues λ1 , λ2 , . . . , λn then
(a) cA has eigenvalues cλ1 , cλ2 , . . . , cλn for any constant c ∈ R.
(b) If A−1 exists, then A−1 has eigenvalues 1/λ1 , 1/λ2 , . . . , 1/λn
m
m
(c) Am has eigenvalues λm
1 , λ2 , . . . , λn for m = 1, 2, 3, . . ..
83. Show that if 0 < θ < π then the matrix
cos θ − sin θ
A=
sin θ cos θ
has no real eigenvalues. Give a geometric interpretation of this fact,
given that the vector Av is obtained from the vector v by a counterclockwise rotation about 0 through an angle θ.
14
Properties of eigenvalues
84. For each of the following matrices:


5 0 0
A = 0 4 1 ,
0 1 4


3 1 1
B = 1 3 1 ;
1 1 3
(a) determine all eigenspaces, Eλ , and state whether the multiplicity
of λ equals the dimension of Eλ ;
(b) check that the determinant equals the product of the eigenvalues,
and the trace is their sum.
85. Answer the following true or false and give reasons.
(a) λ4 + λ2 − 1 cannot be the characteristic polynomial of a 3 × 3
matrix.
(b) An eigenvalue of multiplicity 7 can have an eigenspace of dimension 3.
(c) A real n × n matrix always has a real eigenvalue when n is odd.
(d) If tr(A) = 0 then A cannot be invertible.
(e) A polynomial with real coefficients must have real roots.
(f) If λ is an eigenvalue of an n × n matrix A with eigenvector x,
then for any scalar µ, λ − µ is an eigenvalue of A − µIn with x as
a corresponding eigenvector.
86. (a) Show that λ = 0 is an eigenvalue of A if and only if A is not
invertible.
(b) Without calculation find one eigenvalue and two linearly independent eigenvectors of


1 2 3
A = 1 2 3 .
1 2 3
87. (a) Let A be an n × n matrix. Show that A and its transpose At have
the same characteristic polynomial and hence the same eigenvalues.
(b) Beware! Even though A and At have the same eigenvalues, the
eigenspace for A at an eigenvalue λ need not be equal to the
eigenspace for At at λ. Investigate this phenomenon with the
matrix
1 1
A=
0 1
88. Suppose that for some 3 × 3 matrix A, we have
   
   
1
2
1
2
A 1 = 2 and A 0 = 0
1
2
1
2
15
(a) Give one eigenvalue of A.
 
0

(b) Explain why 1 is an eigenvector for this eigenvalue and hence
0
 
0
find A 1.
0
(c) If det(A)=12, then what is the multiplicity of the eigenvalue from
(a)?
Diagonalisation
89. Consider the matrices
−12 7
A1 =
−7 2


4 2 2
A2 =  2 4 2 
2 2 4


−1
3
9
A3 =  0 −7 −18 .
0
2
5
For each matrix Ai above
(a) Determine the eigenvalues and eigenvectors.
(b) For each eigenvalue, state its multiplicity and give the dimension
of its associated eigenspace.
(c) Hence determine whether the matrix Ai is diagonalisable, stating
the reason for your answer.
(d) For each diagonalisable matrix Ai , determine a matrix P such
that P −1 Ai P = D, where D is a diagonal matrix. What is D?
(For the purposes of this exercise, order your eigenvalues from
smallest to largest i.e. λ1 ≤ λ2 . . . ≤ λn .)
(e) Determine if each matrix Ai is invertible and, where possible, use
your results from (d) to write the inverse matrix A−1
i in the form
P ∆P −1 , where ∆ is a diagonal matrix. (You do not need to find
A−1 if not possible by this method.)
2 3
90. Let A =
.
4 1
1
(a) Verify that v1 =
is an eigenvector of A. What is the corre1
sponding eigenvalue λ1 ?
(b) Use the trace of A to find a second eigenvalue λ2 .
(c) Find the eigenspace for λ2 .
16
(d) Write down an invertible matrix P such that P −1 AP = D, where
D = diag(λ1 , λ2 ).


3 0 0
91. Show that the matrix A = 0 2 0 is not diagonalizable.
0 1 2
92.*
(a) A square matrix A is called nilpotent if Ak = 0 for some integer
k > 1. Show that 0 is the only eigenvalue of a nilpotent matrix.
(b) A square matrix A is called idempotent if A2 = A. Show that
the only possible eigenvalues of A are 0 or 1.
(c) Suppose that A is a diagonalizable matrix such that every eigenvalue of A is 0 or 1. Prove that A is idempotent.
93. Let s be a real number, and let

s
−1
s−1
0
1
1
A=
0
1
1
0 s2 − s s 2 − s

0
0
.
0
2
(a) For which values of s is A diagonalisable?
(b) For s = 0, find an invertible matrix P such that P −1 AP is a
diagonal matrix.
Cayley-Hamilton Theorem
94. Use the Cayley-Hamilton theorem to compute the inverse of the matrix


1 2 5
A = 0 2 −1 .
0 0 1


1 1 0
95. Let A = 1 0 1. Use the Cayley-Hamilton theorem to compute
0 1 1
A3 and A4 .
Dynamical Systems
−5 9
96. Consider the dynamical system with xk+1 = Axk , where A =
.
−3 11
2
Show that one eigenvector of A is (2, 1)T . What is the corresponding
eigenvalue? Hence determine the remaining eigenvalue and its eigenvectors.
What happens to the system over a long period of time?
97.*
17
(a) A square matrix B is called nilpotent if B k = 0 for some integer
k > 1. Show that 0 is the only eigenvalue of a nilpotent matrix.
(b) A square matrix C is called idempotent if C 2 = C. What are the
possible eigenvalues of an idempotent matrix?
98. Find the steady-state probability vectors for the Markov processes
with transition matrices:


1/2 3/4 1/3
3/4 1/3
(a)
(b) 1/2 1/4 1/3
1/4 2/3
0
0 1/3
1−b
a
(c)
b
1−a
(with 0 < a, b < 1)
99. A country is divided into four regions (A, B, C and D). It is found
that each year 18% of the residents in A move to B, 12% move to C
and 20% move to D. Of the residents in B, 30% move to C, 10% move
to D and only 5% move to A. Of the residents in D, 50% move to
A, 25% move to C and 10% move to B; while the residents of C will
either stay where they are or move to A with equal probability. Write
down the transition matrix for this Markov process.
100.* Fibonacci numbers turn up in a huge variety of applications. They
are determined according to the following rules: F0 = 0; F1 = 1;
Fk+1 = Fk + Fk−1 ∀ k ≥ 1.
1
Fk+1
1 1
, k ≥ 0 and x0 =
, xk =
Let A =
0
Fk
1 0
Explain why the sequence {Fk : k = 0, 1, 2...}can be determined from
xk+1 = Axk . Hence show xk+1 = Ak x0 .
Show that Fk =
Find F1000 .
λk1 −λk2
√
5
Show that as k → ∞,
where λ1 , λ2 are eigenvalues of A, (λ1 > λ2 ).
Fk+1
Fk
→
√
1+ 5
2
18
(Golden Ratio) .