4.1 PROBLEMS

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.

1. If the points Pi = (xi; yi); i = 1; 2; 3; 4 form a quadrilateral with ver-

tices in anti{clockwise orientation, prove that the area of the quadri-

lateral equals

1

2

µ¯¯¯¯

x1 x2

y1 y2

¯¯¯¯

+

¯¯¯¯

x2 x3

y2 y3

¯¯¯¯

+

¯¯¯¯

x3 x4

y3 y4

¯¯¯¯

+

¯¯¯¯

x4 x1

y4 y1

¯¯¯¯

:

(This formula generalizes to a simple polygon and is known as the

Surveyor's formula.)

2. Prove that the following identity holds by expressing the left{hand

side as the sum of 8 determinants:

¯¯¯¯¯¯

a + x b + y c + z

x + u y + v z + w

u + a v + b w + c

¯¯¯¯¯¯

= 2

¯¯¯¯¯¯

a b c

x y z

u v w

¯¯¯¯¯¯

:

3. Prove that ¯¯¯¯¯¯

n2 (n + 1)2 (n + 2)2

(n + 1)2 (n + 2)2 (n + 3)2

(n + 2)2 (n + 3)2 (n + 4)2

¯¯¯¯¯¯

= ¡8:

4. Evaluate the following determinants:

(a)

¯¯¯¯¯¯

246 427 327

1014 543 443

¡342 721 621

¯¯¯¯¯¯

(b)

¯¯¯¯¯¯¯¯

1 2 3 4

¡2 1 ¡4 3

3 ¡4 ¡1 2

4 3 ¡2 ¡1

¯¯¯¯¯¯¯¯

.

[Answers: (a) ¡29400000; (b) 900.]

5. Compute the inverse of the matrix

A =

2

4

1 0 ¡2

3 1 4

5 2 ¡3

3

5

by ¯rst computing the adjoint matrix.

[Answer: A¡1 = ¡1

13

2

4 ¡11 ¡4 2

29 7 ¡10

1 ¡2 1

3

5.]

86 CHAPTER 4. DETERMINANTS

6. Prove that the following identities hold:

(i)

¯¯¯¯¯¯

2a 2b b ¡ c

2b 2a a + c

a + b a + b b

¯¯¯¯¯¯

= ¡2(a ¡ b)2(a + b);

(ii)

¯¯¯¯¯¯

b + c b c

c c + a a

b a a + b

¯¯¯¯¯¯

= 2a(b2 + c2):

7. Let Pi = (xi; yi); i = 1; 2; 3. If x1; x2; x3 are distinct, prove that there

is precisely one curve of the form y = ax2 + bx + c passing through

P1; P2 and P3.

8. Let

A =

2

4

1 1 ¡1

2 3 k

1 k 3

3

5 :

Find the values of k for which detA = 0 and hence, or otherwise,

determine the value of k for which the following system has more than

one solution:

x + y ¡ z = 1

2x + 3y + kz = 3

x + ky + 3z = 2:

Solve the system for this value of k and determine the solution for

which x2 + y2 + z2 has least value.

[Answer: k = 2; x = 10=21; y = 13=21; z = 2=21.]

9. By considering the coe±cient determinant, ¯nd all rational numbers a

and b for which the following system has (i) no solutions, (ii) exactly

one solution, (iii) in¯nitely many solutions:

x ¡ 2y + bz = 3

ax + 2z = 2

5x + 2y = 1:

Solve the system in case (iii).

[Answer: (i) ab = 12 and a 6= 3, no solution; ab 6= 12, unique solution;

a = 3; b = 4, in¯nitely many solutions; x = ¡2

3z+ 2

3 ; y = 5

3z¡ 7

6 , with

z arbitrary.]

4.1. PROBLEMS 87

10. Express the determinant of the matrix

B =

2

664

1 1 2 1

1 2 3 4

2 4 7 2t + 6

2 2 6 ¡ t t

3

775

as as polynomial in t and hence determine the rational values of t for

which B¡1 exists.

[Answer: detB = (t ¡ 2)(2t ¡ 1); t 6= 2 and t 6= 1

2 .]

11. If A is a 3 £ 3 matrix over a ¯eld and detA 6= 0, prove that

(i) det (adjA) = (detA)2;

(ii) (adjA)¡1 =

1

detA

A = adj (A¡1):

12. Suppose that A is a real 3 £ 3 matrix such that AtA = I3.

(i) Prove that At(A ¡ I3) = ¡(A ¡ I3)t.

(ii) Prove that detA = §1.

(iii) Use (i) to prove that if detA = 1, then det (A ¡ I3) = 0.

13. If A is a square matrix such that one column is a linear combination of

the remaining columns, prove that detA = 0. Prove that the converse

also holds.

14. Use Cramer's rule to solve the system

¡2x + 3y ¡ z = 1

x + 2y ¡ z = 4

¡2x ¡ y + z = ¡3.

[Answer: x = 2; y = 3; z = 4.]

15. Use remark 4.0.4 to deduce that

detEij = ¡1; detEi(t) = t; detEij(t) = 1

and use theorem 2.5.8 and induction, to prove that

det (BA) = detB det A;

if B is non{singular. Also prove that the formula holds when B is

singular.

88 CHAPTER 4. DETERMINANTS

16. Prove that

¯¯¯¯¯¯¯¯

a + b + c a + b a a

a + b a + b + c a a

a a a + b + c a + b

a a a + b a + b + c

¯¯¯¯¯¯¯¯

= c2(2b+c)(4a+2b+c):

17. Prove that

¯¯¯¯¯¯¯¯

1 + u1 u1 u1 u1

u2 1 + u2 u2 u2

u3 u3 1 + u3 u3

u4 u4 u4 1 + u4

¯¯¯¯¯¯¯¯

= 1 + u1 + u2 + u3 + u4:

18. Let A 2 Mn£n(F). If At = ¡A, prove that detA = 0 if n is odd and

1 + 1 6= 0 in F.

19. Prove that ¯¯¯¯¯¯¯¯

1 1 1 1

r 1 1 1

r r 1 1

r r r 1

¯¯¯¯¯¯¯¯

= (1 ¡ r)3:

20. Express the determinant

¯¯¯¯¯¯

1 a2 ¡ bc a4

1 b2 ¡ ca b4

1 c2 ¡ ab c4

¯¯¯¯¯¯

as the product of one quadratic and four linear factors.

[Answer: (b ¡ a)(c ¡ a)(c ¡ b)(a + b + c)(b2 + bc + c2 + ac + ab + a2).]

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