1. LINEAR EQUATIONS AND MATRICES

1. For each of the following matrices A and vectors b, use Gaussian elimination to find the general

solution of the system Ax = b.

a) A =

1 −4 −1−1 3 2

2 −9 2

, b =

12−7

35

;

b) A =

0 −3 1−1 −1 2

2 −7 1

, b =

−10

7

;

c) A =

1 −3 4 −13 −1 0 5

−1 3 −2 5

, b =

111

1

;

d) A =

3 4 2 0 −3−1 0 −2 4 1

2 7 −3 −5 −2

, b =

37

−26

.

2. Consider a system of linear equations Ax = b, where A is an m× n matrix.

a) By considering the possible shapes that the echelon form of A may take, show that if m < n

then the system cannot have a unique solution.

b) Show that if m > n there must be some b for which the system has no solution.

3. You are given some systems of linear equations in three variables x1, x2 and x3. Write down

the general solution of each system. (You should not need to do any working!)

a) x1 + 2x2 = 4, x2 + 3x3 = 7;

b) x1 + 2x2 = 4, x3 = 7;

c) x1 + 2x2 = 4, x2 = 7;

d) x1 + 2x2 = 4.

4. Find conditions on b1, b2, b3 such that the triple of equations

2x1 + 3x2 + 4x3 = b1 , 3x1 + 4x2 + 5x3 = b2 , 4x1 + 5x2 + 6x3 = b3

has a solution. Find a formula giving all solutions.

5. For each of the following matrices A, find conditions (if any) on b1, b2, . . . , bn such that the

system Ax = b, where b = (b1, b2, . . . , bn)

T , has a solution.

1 −3 32 −5 4

2 −9 12

;

1 −3 2−3 14 −8

−1 −7 3

;

−1 1 3

3 3 −6

−1 3 −1

0 −2 1

.

6. Show that the system of three equations x+ y+2z = a, x+ z = b, 2x+ y+3z = c is consistent

if and only if c = a+ b.

1

7. For which values of c does the system

x+ 2y + cz = 1

−x+ cy − z = 0

cx− 4y + cz = −1

have (a) no solutions; (b) infinitely many solutions; (c) a unique solution? Find the solutions

when they exist.

8. Let

A =

2 −3 43 2 −2

1 −1 3

, B =

−2 13 4

−1 5

, C =

−3 21 −4

6 2

, D =

(

2 3 1

1 −2 3

)

.

Evaluate each of the following expressions if it exists, or give a reason why it does not exist.

Here I denotes an identity matrix of appropriate size.

3A ; −2B ; A+B ; B + C ; A+ 3I ; B + 3I ; BD ; DB ;

AB ; BC ; A2 ; (BD)2 ; AT ; BT ; BTB ; BBT .

9. Write down the inverses of the matrices A =

(

4 5

3 4

)

and B =

(

1 4

9 16

)

.

10. Find the inverse (if it exists) of each of the following matrices.

A =

1 3 −20 1 −2

0 0 1

, B =

0 2 01 2 3

−1 4 −2

, C =

1 2 32 3 4

3 4 5

, D =

1 4 12 3 1

1 −7 −2

,

E =

0

7

2 −1

2 1 −4

1

2 −3 0

, F =

3 −3 −23 −4 −2

−4 3 3

, G =

5 −6 1 5

−3 5 −1 4

2 −2 1 2

1 −1 0 1

.

11. a) Let B be a square matrix. Show that BBT is symmetric; and B +BT is symmetric; and

B −BT is skew–symmetric.

b) Show that every square matrix can be written as the sum of a symmetric matrix and a

skew–symmetric matrix.

12. Find a 2× 2 matrix A, other than A = ±I, such that A2 = I.

13. Suppose that K is a skew–symmetric matrix and I +K is non–singular. Prove that the matrix

(I +K)−1(I −K) is orthogonal.

14. It is given that A,B,C are n×n invertible matrices, and that A is symmetric, B skew–symmetric

and C orthogonal. Simplify the following expressions.

a) A−1(CB3A)TC;

b) A(BA)TB−1C6(BC7)TB;

c) C(BC)−1BTA5(B2A4)−1C(A−1BC)T .

2

2. VECTOR SPACES

1. Are the following sets vector spaces? Give reasons for your answers.

a) {x ∈ R2 | x21 = x

3

2 };

b) {x ∈ R3 | 2x1 − 3x2 + 3x3 = 0 };

c) {x ∈ R3 | 2x1 − 3x2 + 3x3 = 13 };

d) { t1u1 + t2u2 | t1, t2 ∈ R }, where u1 and u2 are fixed elements of R

3;

e) {x ∈ R3 | 2x1 + 3x2 − 4x3 ≤ 6 };

f) {x ∈ R4 | 2x1 + 3x2 − 4x4 = 0 }.

2. Are the following sets vector spaces? Give reasons.

a) { p ∈ P4 | p(1) = 0 and p(−1) = 0 };

b) { p ∈ P3 | xp

′(x) = 2p(x) for all x ∈ R };

c) { p ∈ P9 | p has degree 4 or more }.

3. Let V = C[0, 2] be the vector space of continuous real–valued functions on [0, 2], where addition

and scalar multiplication are defined by

(f + g)(t) = f(t) + g(t) and (λf)(t) = λf(t)

for all f, g ∈ V and all λ ∈ R. It is given that V is a vector space under these operations. Define

W1 =

{

f ∈ V

∣∣∣∣

∫ 2

0

f(t) dt = 1

}

and W2 =

{

f ∈ V

∣∣∣∣

∫ 2

0

f(t) dt = f(1)

}

.

Prove that W1 is not a vector space, but W2 is.

4. Let V be the set of all infinite sequences {αn } = (α0, α1, α2, . . . ) of elements of R. Define

addition and scalar multiplication of sequences by

{αn }+ {βn } = {αn + βn } and λ{αn } = {λαn } .

Prove that

a) V is a vector space over R;

b) the set

{

{αn } ∈ V

∣∣ αn+2 = αn+1 + αn for all n ≥ 0} of all Fibonacci sequences is a sub-

space of V ;

c) the set

{

{αn } ∈ V

∣∣ {αn } converges } of all convergent sequences is a subspace of V .

5. Are the following sets of vectors linearly independent? Don’t do any more calculation than you

have to!

a) { (1, 2, 0), (4, 5, 1), (1,−1, 7) } in R3.

b) { (1, 2, 3), (2, 3, 4), (3, 4, 5) } in R3.

c) { (4,−1, 1), (7, 2, 3), (−1, 4,−2), (9, 2,−1) } in R3.

d) { (3, 1, 4, 1, 5, 9), (2, 6, 5, 3, 5, 8) } in R6.

e) { (2, 1, 3,−2), (−1, 0, 1, 7), (−6,−3,−9, 6) } in R4.

3

6. Is the set

12

0

,

45

1

,

1−1

7

a linearly independent subset of R3? Comment on your answer.

7. Are the following sets of vectors linearly independent?

a) { 1 + 2t, 4 + 5t+ t2, 1− t+ 7t2 } in P2.

b) {−1 + t, 2 + t2, −3 + t+ t2, −1− t− t3, 2 + 3t2 + t3 } in P3.

c)

{(

1 2

0 1

)

,

(

2 0

1 3

)

,

(

1 1

4 0

)}

in M2,2.

8. Let u, v, w be elements of a vector space V , and suppose that the set S = {u, v } is linearly

independent. Prove that {u, v, w } is linearly dependent if and only if w is in span(S).

9. Some linear independence problems using more advanced techniques.

a) Show that { cos t, sin t, t cos t, t sin t } is independent by considering a possible identity

λ1 cos t + λ2 sin t + λ3t cos t + λ4t sin t = 0 and evaluating the left hand side at carefully

chosen values of t.

b) Use differentiation to show that the set { 1, ex, e2x } is linearly independent.

c) Let f(t) = λ1 cos t+ λ2 sin t+ λ3 cos 2t+ λ4 sin 2t. By evaluating

∫ 2pi

0

f(t) cos t dt ,

∫ 2pi

0

f(t) sin t dt ,

∫ 2pi

0

f(t) cos 2t dt ,

∫ 2pi

0

f(t) sin 2t dt

show that the set of functions { cos t, sin t, cos 2t, sin 2t } is linearly independent.

10. Let V be the vector space of all twice differentiable real–valued functions defined on R. If f, g

are in V then the determinant

Wf,g(t) =

∣∣∣∣f(t) g(t)f ′(t) g′(t)

∣∣∣∣ = f(t)g′(t)− f ′(t)g(t)

is called the Wronskian of f and g; it is significant in the theory of linear ordinary differential

equations.

a) Prove that if { f, g } is a linearly dependent set in V then Wf,g(t) = 0 for all t.

b) Show that { cos t, sin t } is a linearly independent set.

11. Let S = { (1,−1, 3), (−1, 3,−7), (2, 1, 0) }. Do the vectors u = (5, 1, 3) and v = (2, 3, 6) belong

to span(S)?

12. Let S = { 1 − t + 3t2, 1 − t2, 2 + t + 5t2 }. Does the polynomial p(t) = 1 + t + t2 belong to

span(S)?

13. Let S = { 1− t+ 3t2, 1− t2, 2 + t+ 5t2 }. Does the polynomial p(t) = 12941− 7696t+50114t2

belong to span(S)? Suggestion. Look back at the previous problem before you do any annoying

calculations.

14. In the following questions, try to do as little calculation as possible. Give reasons for all your

answers.

a) Is the set S in problem 11 a spanning set for R3?

4

b) Is the set S in problem 12 a spanning set for P2?

c) Is S = { (1, 3,−1, 5), (7, 2, 2,−6) } a spanning set for R4?

d) Is S =

{(

1 2

0 1

)

,

(

2 0

1 3

)

,

(

1 1

4 0

)

,

(

0 1

1 0

)}

a spanning set for M2,2?

15. Are the four polynomials p1(t) = −1 − 3t

2 + t3, p2(t) = −1 + 2t + t

2, p3(t) = 1 − 2t + t

3,

p4(t) = 1 − 6t − 4t

2 + 3t3 linearly dependent in P3? If so, write p4 as a linear combination of

the other three. Otherwise, write p(t) = −2 + 6t + 2t2 − t3 as a linear combination of all four

polynomials.

16. Are the following sets bases for R3?

a) S = { (1, 0, 4), (−1, 1, 3) };

b) S = { (1, 0, 4), (−1, 1, 3), (2,−3,−6) };

c) S = { (1, 0, 4), (−1, 1, 3), (2,−3,−6), (5, 1,−8) };

d) S = { (−1, 1, 2), (0, 3, 1), (3, 0,−5) }.

17. Are the following sets bases for P3?

a) S = { 1 + 6t, −1 + t+ t2, 2 + 3t2 − t3 };

b) S = {−1 + 2t, 1− 3t+ t3, t+ t2 − 2t3, 1 + 2t2 + t3 }.

18. Find bases for the following vector spaces, and state their dimension.

a) {x ∈ R3 | x1 + 3x2 − 4x3 = 0 };

b) {x ∈ R4 | x1 + 3x2 + x3 − x4 = 0 };

c) { p ∈ P2 | p(2) = 0 and p(−1) = 0 };

d) { p ∈ P2 | p(1) + p(3) = 0 }.

19. Find the coordinate vector of

a) b = (9,−5, 8) with respect to the basis B = { (1, 0, 4), (−1, 1, 3), (2,−3,−6) } of R3;

b) p(t) = 8− 9t+ 3t2 + 5t3 with respect to the basis

B = {−1 + 2t, 1− 3t+ t3, t+ t2 − 2t3, 1 + 2t2 + t3 }

of P3;

c) b = (b1, b2) with respect to the basis B = { (2, 3), (3, 5) } of R

2.

20. Show that the set B = { t2 + t3, 1+ t+ t2+ t3, t, 1+ t2+2t3 } is linearly independent. Explain

without further calculation why this set forms a basis for P3, and obtain the coordinates of

1 + 2t+ 3t2 + 4t3 with respect to B.

21. Let p1(t) =

1

2(t − 2)(t − 3), p2(t) = −(t − 1)(t − 3) and p3(t) =

1

2(t − 1)(t − 2). Let p(t) be a

polynomial of any degree. Prove that if the equation

p = λ1p1 + λ2p2 + λ3p3

has a solution at all, it has the unique solution λ1 = p(1), λ2 = p(2), λ3 = p(3). Deduce that

B = { p1, p2, p3 } is a basis for P2, and that the coordinate vector with respect to B of any p in

P2 is

(

p(1), p(2), p(3)

)T

.

5

22. Find conditions on b1, b2, b3 such that the vector b = (b1, b2, b3)

T is in the column space of the

matrix

A =

1 −3 32 −5 4

2 −9 12

.

23. Find bases for the kernels (nullspaces) and column spaces of the following matrices. Hence

obtain the nullity and rank of each matrix.

A =

(

2 3 1

1 −2 −3

)

, B =

−1 2 13 −5 −4

−2 5 8

, C =

1 3 −1 1−2 −6 2 −2

−1 −3 1 −1

,

D =

1 0 −1 3 13 −1 −1 7 2

1 −3 5 −2 −6

, E =

1 3 1

2 5 0

1 −2 −9

−1 1 7

.

3. LINEAR TRANSFORMATIONS

1. In solving the following problems, give clearly written and logically complete arguments; and

make sure that the differences between the two problems are clear.

a) Prove that the set S = {x ∈ R4 | 5x1 + x3 − 2x4 = 0 } is a subspace of R

4.

b) Prove that the function T : R4 → R defined by T (x) = 5x1 + x3 − 2x4 is a linear transfor-

mation.

2. Are the following functions linear? Prove your answers.

a) T : R2 → R2, T (x1, x2) = (2x1, x1 − x2);

b) T : R2 → R2, T (x1, x2) = (x1 + 1, x2);

c) T : P2 → P2, T (a+ bt+ ct

2) = (a+ c)− (c+ b)t+ (a+ b+ c)t2;

d) T : P → P, T (p) = p+2p′+3p′′, where P is the vector space of polynomials (of any degree);

e) T : P2 → P2, T

(

p(t)

)

= p(t− 2);

f) det :M2,2 → R, det

(

a b

c d

)

= ad− bc;

g) F :M2,2 →M2,2, F (X) = X

T , the transpose of X.

3. Let T : R2 → R2 be a linear transformation. Find a formula for T (x1, x2), given that

a) T (1, 0) = (3, 4) and T (0, 1) = (4, 9);

b) T (4, 7) = (3, 4) and T (3, 5) = (4, 9);

c) T (5, 7) = (3, 4) and T (2, 7) = (2, 5).

4. For each linear transformation in question 2 having finite–dimensional domain and codomain,

find the matrix of the transformation with respect to standard bases.

5. A function T : P2 → P2 is defined by T

(

p(t)

)

= tp′(t).

a) Show that T is linear.

b) Find the matrix of T with respect to the basis B = { 1, 1 + t, t2 } of P2.

6

6. Let T1 and T2 be linear mappings from P1 to R

2 defined by

T1(a+ bt) = (2a+ 3b, a− b) and T2(a+ bt) = (a− b, a+ b) .

a) Find the matrices of T1 and T2 relative to the standard bases { 1, t } in P1 and { (1, 0), (0, 1) }

in R2.

b) Find the matrices of T1 and T2 relative to the basis B = { 1 + t, 2− t } in P1 and the basis

C = { (2, 3), (1, 2) } in R2.

7. Given that the linear function T : R2 → R2 has matrix A with respect to the standard basis of

R

2, find the matrix of T with respect to the basis B, if

a) A =

(

−2 1

5 2

)

and B = { (1, 5), (1, 6) };

b) A =

(

4 9

1 1

)

and B = { (1,−1), (3, 2) };

c) A =

(

6 −1

12 −1

)

and B = { (1, 4), (1, 3) }.

8. If T : R2 → R2 has matrix

A =

(

4 3

2 1

)

with respect to the basis B = { (1, 3), (3, 7) } of R2, find the matrix of T with respect to the

standard basis.

9. Let v1 = (1,−2, 0), v2 = (0,−1, 1), v3 = (1, 0,−1); suppose that T : R

3 → R3 is a linear

transformation and that

T (v1) = 5v1 + v2 , T (v2) = 5v2 + v3 , T (v3) = 5v3 .

a) Write down the matrix of T with respect to the basis consisting of the vectors v3, v2, v1,

in that order.

b) Find the matrix of T with respect to the standard basis for R3. Comment. This kind of

problem will be exceedingly important in Chapter 10.

10. Find bases for the kernels and images of the following linear transformations. Hence obtain the

nullity and rank of each transformation.

a) T : R4 → R3, T (x) =

1 −1 1 20 1 3 1

0 0 0 6

x;

b) T : R4 → R3, T (x) =

1 2 −1 05 9 −8 1

3 8 3 −2

x;

c) T : P4 → R

2, T (p) =

(

p(0), p′(0)

)

;

d) F :M3,3 →M3,3, F (X) = X −X

T where XT denotes the transpose of X.

11. The mapping T : P3000 → P3000 is given by T (p) = p

(500), the 500th derivative of p. Without

doing any calculation, write down the kernel, the nullity and the rank of T .

7

4. LEAST SQUARES

1. For the vectors

u = (−1,−4, 1) , v = (1,−1,−6) , w = (1, 1, 2)

find a unit vector in the direction of u, decide whether or not u and v are perpendicular, find

the angle between u and w, find a vector perpendicular to both v and w, and find the projection

of v onto w. Repeat for the vectors

u = (2,−1,−4) , v = (3, 2, 1) , w = (−1, 2, 2) .

2. Find a basis for the orthogonal complement W⊥ of

a) W = {x ∈ R3 | 5x1 − 7x2 − x3 = 0 } in R

3;

b) W = span{ (2,−1, 2), (3, 1,−1) } in R3;

c) W = span{ (3, 4,−1) } in R3;

d) W = span{ (1,−1, 4, 2), (2,−1, 7, 5) } in R4;

e) W = {x ∈ R4 | −x1 + 3x2 + 7x3 − 2x4 = 0 } in R

4.

3. Let W be the subspace of R3 spanned by (1, 2, 2) and (2, 1,−2). Find

a) the projection of x = (2, 9,−4) onto W ;

b) the projection of x = (2, 9,−4) onto W⊥.

4. Find the projection of v onto W = span{w1,w2, . . . }, if

a) v = (4, 7, 8) and w1 = (−4, 1, 1), w2 = (3,−1, 0) in R

3;

b) v = (1, 9,−3, 7) and w1 = (1, 2, 1, 0), w2 = (−1,−1, 1, 2) in R

4;

c) v = (4, 8, 3,−2) and w1 = (2,−1, 1, 0), w2 = (0, 1,−1, 1), w3 = (1, 0, 1, 3) in R

4.

5. Confirm that B is an orthonormal set, and find the projection of v onto spanB, if

a) B = { 1√

6

(−1, 1, 2), 1√

3

(1,−1, 1) } and v = (8, 4, 5);

b) B = { 1√

7

(1, 2,−1, 1), 1√

11

(0, 1, 3, 1) } and v = (2,−6, 7,−4).

6. Find the matrix of the linear transformation T : R3 → R3, where T (v) is the projection of v

onto W = span{ (−4, 1, 1), (3,−1, 0) }. Use this matrix to check your answer to problem 4(a).

Do the same for T : R4 → R4, the projection onto W = span{ 1√

7

(1, 2,−1, 1), 1√

11

(0, 1, 3, 1) },

and hence check your answer to 5(b).

7. Use the Gram–Schmidt process to find orthonormal bases for the spaces

a) span{ (5, 12), (−4, 6) };

b) span{ (−2, 1,−2), (1, 4,−8) };

c) span{ (2,−3, 6), (1, 1,−1) };

d) span{ (0, 1,−1, 2), (1, 0, 4,−4), (−5, 5,−7, 6) };

e) span{ (1, 2, 1,−1), (0,−6,−5, 4), (−2, 7, 6, 4) }.

8. Find a QR factorisation for each of the matrices

A =

(

5 −4

12 6

)

, B =

−2 11 4

−2 −8

, C =

1 3 2

5 4 4

1 −2 −1

−3 1 3

, D =

−1 7 118 7 11

4 8 1

.

8

9. Find the least squares solution of Ax = b, where

a) A =

1 1−1 1

1 2

and b =

113

4

;

b) A =

1 2

0 1

−1 3

1 −1

and b =

−1

4

2

1

.

10. Find the line y = a + bx that best fits in the least squares sense the three points (1, 1), (3, 2)

and (4, 6).

11. For the points (−1, 7), (0, 4), (1,−2), (2,−6),

a) find the line y = a+ bx that is a best fit in the least squares sense;

b) find the quadratic y = a+ bx+ cx2 that is a best fit in the least squares sense.

12. Let P,Q,R be the three points (1, 1), (2, 1) and (3, 1) respectively.

a) Find the line y = a+ bx that best fits P,Q and R.

b) Find the line y = a+ bx that best fits P,Q,R and the origin.

c) Find the line y = bx through the origin that best fits P,Q and R. Why is this not the same

as the line in (b)?

13. A farmer fertilises four fields with different amounts x of fertiliser and gets different yields y.

The four yields are 12 , 1,

5

2 and 3 for fertiliser amounts 0, 1, 2, and 3 respectively. Find the line

of best fit y = a + bx through these points. Hence estimate the yield the farmer would obtain

by using 5 units of fertiliser. Comments?

14. The voltage V of a discharging battery after 0, 12 , 1 and 1

1

2 minutes is measured as 3.5, 2, 1.5

and 1 volts respectively. Find the quadratic of best fit for the voltage as a function of time.

15. Find values of a, b, c, d such that the curve y = a + bx + cx−1 + d cos x best fits, in the least

squares sense, the points (1, 3), (2, 1), (3, 7), (4, 9) and (6,−5). You will probably want to do

the calculations on Maple; you could also use Maple to plot your curve and compare it with the

given points.

16. Let v1 = (0, 1, 2, 2), v2 = (−6,−1,−2, 7), v3 = (2, 0,−9, 6) be three vectors in R

4, and write

W = span{v1,v2,v3 }.

a) Using the Gram–Schmidt process, give an orthonormal basis for W .

b) Hence find a QR factorisation of the matrix

A =

0 −6 2

1 −1 0

2 −2 −9

2 7 6

.

c) Hence, or otherwise, show that the plane ax+ by + cz − w = 0 that best fits (in the least

squares sense) the four points

(x, y, z, w) = (0,−6, 2,−2), (1,−1, 0, 2), (2,−2,−9,−3), (2, 7, 6, 5)

is given by a = 59 , b =

1

3 , c =

1

3 .

9

17. Which of the following functions are inner products on R2? Give reasons.

a) 〈x,y〉 = x1y2 + y1x2;

b) 〈x,y〉 = x1y1 + 3x2y2;

c) 〈x,y〉 = x1y2 + 2y1x2;

d) 〈x,y〉 = x1y1 − 3x2y2.

18. For a real inner product space

a) prove that ‖u+v‖2+‖u−v‖2 = 2‖u‖2+2‖v‖2. Why is this known as the parallelogram

identity?

b) prove the identity 4〈u,v〉 = ‖u+ v‖2 − ‖u− v‖2.

c) prove Pythagoras’ Theorem: 〈u,v〉 = 0 if and only if ‖u‖2 + ‖v‖2 = ‖u+ v‖2.

19. (Harder.) Let u and v be two non-zero vectors in Rn. Show that the vector

w =

1

(‖u‖+ ‖v‖)

(‖u‖v + ‖v‖u)

bisects the angle between u and v.

20. By applying the Gram–Schmidt process to the standard basis of P3, find an orthonormal basis

for P3 with respect to the inner product

〈p, q〉 =

∫ 1

−1

p(t)q(t) dt .

The polynomials you have found are constant multiples of the very important Legendre poly-

nomials.

21. Let V be a finite–dimensional inner product space and let W be a subspace of V . Define

W⊥ = {v ∈ V | 〈v,w〉 = 0 for all w ∈W } .

Show that

a) W⊥ is a subspace of V ;

b) W ∩W⊥ = {0 };

c) for each v ∈ V there are unique vectors w1 ∈W and w2 ∈W

⊥ such that v = w1 +w2;

d) W⊥⊥ =W ;

e) dimW + dimW⊥ = dimV .

22. Find the reflection of

a) v = (−8, 12,−7) in the plane perpendicular to d = (3,−1, 3)T ;

b) v = (5, 7,−3) in the plane x1 + 4x2 − x3 = 0;

c) v = (1,−2, 6,−3) in the hyperplane perpendicular to d = (1, 1,−1, 4)T .

23. Find the matrix of a reflection that interchanges the vectors v and w, or explain why no such

reflection exists, when

a) v = (12, 3,−5) and w = (13, 0,−3);

b) v = (−1, 0, 0, 7) and w = (−3, 2,−1, 6);

c) v = (0,−9, 0) and w = (8, 0,−4);

d) v = (3,−1, 2, 4) and w = (5, 0,−1, 2).

10

5. DETERMINANTS

1. Using the simplest method you can find, evaluate the determinants of the matrices

A =

(

5 7

−1 2

)

, B =

(

9 11

2 −10

)

, C =

0 0 32 1 −4

9 −1 0

, D =

1 −3 1−3 7 −8

8 −2 1

E =

1 4 −2−2 −5 5

1 −8 −6

, F =

0 6 −11 5 2

−7 1 −9

, G =

16 −9 43152 97 −346

−160 90 −430

,

H =

−1 3 1 2

2 −6 4 5

3 −4 −4 −5

6 −8 −2 9

, J =

1 −1 2 1

5 −2 9 7

−4 7 4 −3

3 −9 −5 0

, K =

−1 2 −1 −1

4 −3 4 4

−3 6 −4 0

−3 1 −1 −1

.

2. Which of the matrices in question 1 are invertible? (Don’t do any more calculation!)

3. Let A be a 5× 5 matrix with determinant 6. Let B be the matrix that results from multiplying

the matrix A by 2. Let C be the matrix obtained from B by adding twice the first row of B

to the third row of B. Finally let D be the matrix obtained by swapping the first and fourth

columns of C and dividing the second row by 3. What are the determinants of B, C and D?

4. a) Explain why the determinant

det

x

3 8 1

x 2 1

1 1 1

,

when expanded, is a polynomial in x, and give the degree of this polynomial.

b) Find by inspection two values of x for which the determinant is zero.

c) Hence find all the roots of the polynomial.

5. Let a, b and c be three non–collinear points in R3. Prove that

det

1 x1 x2 x3

1 a1 a2 a3

1 b1 b2 b3

1 c1 c2 c3

= 0

is the equation of the plane through these three points. Comment. It is actually quite difficult

to give a complete answer to this question. Try this: (i) explain why the determinant, when

expanded, has more or less the right form to be the equation of a plane, and why the points

a, b and c satisfy this equation; (ii) identify any possible case in which the equation is not in

fact that of a plane; (iii) prove that this case can only arise if the given points are collinear.

6. Find all values of x such that ∣∣∣∣∣∣∣∣

x a b c

a x b c

a b x c

a b c x

∣∣∣∣∣∣∣∣

= 0 .

7. Show that if A and B are invertible n×n matrices then det

(

(AT )7B−15A29(BT )11A−36B4

)

= 1.

11

8. Give an example of 2× 2 matrices A and B such that det(A+B) 6= detA+ detB. What does

this tell you about the function det :M2,2 → R?

9. For any integer n ≥ 2 and any real number a1, a2, . . . , an, the expression

Vn = det

1 a1 a

2

1 · · · a

n−1

1

1 a2 a

2

2 · · · a

n−1

2

...

...

...

...

1 an a

2

n · · · a

n−1

n

is called a Vandermonde determinant.

a) Show that V3 = (a2 − a1)(a3 − a1)(a3 − a2).

b) (Harder.) Show that in general

Vn =

∏

1≤i

10. Let

A =

(

−2 3 −1

1 4 5

)

and B =

−3 21 −4

6 2

.

Calculate the determinant of the 2 × 2 matrix AB and explain without calculation why the

determinant of the 3× 3 matrix BA is zero.

6. EIGENVALUES, EIGENVECTORS, DIAGONALISATION

1. Find all the eigenvalues, and the eigenvectors corresponding to each eigenvalue, for each of the

matrices

A =

(

1 2

−1 4

)

, B =

(

2 −2

−2 5

)

, C =

(

2 1

−1 0

)

, D =

(

−2 4

−5 7

)

, E =

(

1 4

−1 1

)

,

F =

2 1 11 1 0

1 0 1

, G =

2 0 00 3 1

0 1 3

, H =

3 1 28 7 12

−2 −2 −2

.

2. For each of the matrices in question 1, either diagonalise the matrix or explain why it is not

diagonalisable. If the matrix is diagonalisable, find a formula for its nth power.

3. For each eigenvalue of each matrix in question 1, write down the algebraic multiplicity, the

corresponding eigenspace and the geometric multiplicity.

4. In each of the following cases explain, giving reasons, whether a matrix A exists satisfying the

given conditions:

a) A is a 3× 3 matrix which has eigenvalue λ = 5 only;

b) A is a 4 × 4 matrix having eigenvalues λ = 8, with algebraic multiplicity 2 and geometric

multiplicity 1; and λ = −4, with algebraic multiplicity 1; and λ = 3, with algebraic

multiplicity 1;

c) A is a 6 × 6 matrix having eigenvalues λ = 1, with algebraic multiplicity 3 and geometric

multiplicity 2; and λ = 2, with algebraic multiplicity 2 and geometric multiplicity 2; and

λ = 3, with algebraic multiplicity 1 and geometric multiplicity 2;

12

d) A is a 7×7 matrix which has eigenvalues λ = 3 with algebraic multiplicity 2 and geometric

multiplicity 2; and λ = −1 with algebraic multiplicity 6 and geometric multiplicity 4;

e) A is a 2× 2 matrix with an eigenvalue λ = −1 and corresponding eigenvector v = 0 only;

f) A is a real 2× 2 matrix with eigenvalues 1 + 2i and 3 + 4i;

g) A is a real 2× 2 matrix with eigenvalues 5 + 6i and 5− 6i;

h) A is a real 5× 5 matrix with eigenvalues 1 + 2i and 3 + 4i, and possibly others.

5. For each of the following matrices, use the given additional information to find all eigenvalues and

eigenvectors without calculating the characteristic polynomial. Also write down the algebraic

and geometric multiplicities of each eigenvalue.

a) A =

(

0 −1

−2 1

)

, given that −1 is an eigenvalue;

b) B =

(

2 5

6 1

)

, given that 7 is an eigenvalue;

c) C =

2 −5 −5−4 8 4

4 −11 −7

, given that 2 and −3 are eigenvalues;

d) D =

1 4 22 1 −2

−3 4 6

, given that

10

1

and

2−1

2

are eigenvectors;

e) E =

6 −2 12 1 2

3 −6 8

, given that there is only one eigenvalue;

f) F =

4 1 −1 2

2 5 −2 4

4 4 −1 8

−1 −1 1 1

, given that

−2

0

0

1

is an eigenvector.

6. Find the characteristic polynomials of the matrices

C3 =

0 0 −a01 0 −a1

0 1 −a2

and C4 =

0 0 0 −a0

1 0 0 −a1

0 1 0 −a2

0 0 1 −a3

.

7. Let

M =

2 −5 5−1 −12 13

−1 −19 20

and x =

12

3

and P =

1 0 02 1 0

3 0 1

.

Show that Mx = 7x, and hence write down with a minimum of calculation P−1MP .

8. Let V be a vector space and {v1,v2,v3 } a basis for V . Let T be a linear map from V to V

such that

T (v1) = 2v1 + v2 + v3 , T (v2) = 2v2 , T (v3) = v2 + v3 .

Is there a basis B for V such that the matrix of T with respect to B is diagonal? Explain.

9. a) Show that if A and B are matrices so that AB and BA are both defined then AB and BA

have the same trace. Hence show that S−1MS and M have the same trace.

13

b) Show that M =

12 13 1417 18 19

23 24 70

and N =

30 41 4221 35 −23

38 11 36

are not similar.

10. (You may want to use computer assistance for the calculations in this question.) On the island of

Kerguelen three species A,B,C of feral pigs are in mutual conflict, which is broken off each spring

for breeding. The numbers a′, b′, c′ of each species on 1 December of any year is determined in

terms of the numbers a, b, c of the species on 1 December of the previous year by the formula

a′ = 75a−

1

10b−

1

8c , b

′ = −15a+

13

10b−

1

8c , c

′ = −15a−

1

5b+

5

4c .

Find the population ratio corresponding to a stable population.

11. The Fibonacci numbers {fn}n≥0 are defined by

fn+2 = fn+1 + fn for n ≥ 2 with f0 = 0, f1 = 1 .

In this question we shall use matrix methods to investigate properties of the Fibonacci numbers.

a) If A =

(

0 1

1 1

)

, show that An =

(

fn−1 fn

fn fn+1

)

for all n ≥ 1.

b) Show that fn+1fn−1 = f2n + (−1)

n.

c) By diagonalising the matrix A in (a) find a formula for fn.

12. Find all the eigenvalues and eigenspaces of the following linear maps:

a) T defined on the set of all polynomials by T (p(t)) = p(−t);

b) T defined on the space of all differentiable functions by T (y) =

1

x

dy

dx

.

14