1.6 PROBLEMS

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1. Which of the following matrices of rationals is in reduced row{echelon

form?

(a)

2

4

1 0 0 0 ¡3

0 0 1 0 4

0 0 0 1 2

3

5 (b)

2

4

0 1 0 0 5

0 0 1 0 ¡4

0 0 0 ¡1 3

3

5 (c)

2

4

0 1 0 0

0 0 1 0

0 1 0 ¡2

3

5

(d)

2

664

0 1 0 0 2

0 0 0 0 ¡1

0 0 0 1 4

0 0 0 0 0

3

775

(e)

2

664

1 2 0 0 0

0 0 1 0 0

0 0 0 0 1

0 0 0 0 0

3

775

(f)

2

664

0 0 0 0

0 0 1 2

0 0 0 1

0 0 0 0

3

775

(g)

2

664

1 0 0 0 1

0 1 0 0 2

0 0 0 1 ¡1

0 0 0 0 0

3

775

. [Answers: (a), (e), (g)]

2. Find reduced row{echelon forms which are row{equivalent to the following

matrices:

(a)

·

0 0 0

2 4 0

¸

(b)

·

0 1 3

1 2 4

¸

(c)

2

4

1 1 1

1 1 0

1 0 0

3

5 (d)

2

4

2 0 0

0 0 0

¡4 0 0

3

5 :

18 CHAPTER 1. LINEAR EQUATIONS

[Answers:

(a)

·

1 2 0

0 0 0

¸

(b)

·

1 0 ¡2

0 1 3

¸

(c)

2

4

1 0 0

0 1 0

0 0 1

3

5 (d)

2

4

1 0 0

0 0 0

0 0 0

3

5.]

3. Solve the following systems of linear equations by reducing the augmented

matrix to reduced row{echelon form:

(a) x + y + z = 2 (b) x1 + x2 ¡ x3 + 2x4 = 10

2x + 3y ¡ z = 8 3x1 ¡ x2 + 7x3 + 4x4 = 1

x ¡ y ¡ z = ¡8 ¡5x1 + 3x2 ¡ 15x3 ¡ 6x4 = 9

(c) 3x ¡ y + 7z = 0 (d) 2x2 + 3x3 ¡ 4x4 = 1

2x ¡ y + 4z = 1

2 2x3 + 3x4 = 4

x ¡ y + z = 1 2x1 + 2x2 ¡ 5x3 + 2x4 = 4

6x ¡ 4y + 10z = 3 2x1 ¡ 6x3 + 9x4 = 7

[Answers: (a) x = ¡3; y = 19

4 ; z = 1

4 ; (b) inconsistent;

(c) x = ¡1

2 ¡ 3z; y = ¡3

2 ¡ 2z, with z arbitrary;

(d) x1 = 19

2 ¡ 9x4; x2 = ¡5

2 + 17

4 x4; x3 = 2 ¡ 3

2x4, with x4 arbitrary.]

4. Show that the following system is consistent if and only if c = 2a ¡ 3b

and solve the system in this case.

2x ¡ y + 3z = a

3x + y ¡ 5z = b

¡5x ¡ 5y + 21z = c:

[Answer: x = a+b

5 + 2

5z; y = ¡3a+2b

5 + 19

5 z, with z arbitrary.]

5. Find the value of t for which the following system is consistent and solve

the system for this value of t.

x + y = 1

tx + y = t

(1 + t)x + 2y = 3:

[Answer: t = 2; x = 1; y = 0.]

1.6. PROBLEMS 19

6. Solve the homogeneous system

¡3x1 + x2 + x3 + x4 = 0

x1 ¡ 3x2 + x3 + x4 = 0

x1 + x2 ¡ 3x3 + x4 = 0

x1 + x2 + x3 ¡ 3x4 = 0:

[Answer: x1 = x2 = x3 = x4, with x4 arbitrary.]

7. For which rational numbers ¸ does the homogeneous system

x + (¸ ¡ 3)y = 0

(¸ ¡ 3)x + y = 0

have a non{trivial solution?

[Answer: ¸ = 2; 4.]

8. Solve the homogeneous system

3x1 + x2 + x3 + x4 = 0

5x1 ¡ x2 + x3 ¡ x4 = 0:

[Answer: x1 = ¡1

4x3; x2 = ¡1

4x3 ¡ x4, with x3 and x4 arbitrary.]

9. Let A be the coe±cient matrix of the following homogeneous system of

n equations in n unknowns:

(1 ¡ n)x1 + x2 + ¢ ¢ ¢ + xn = 0

x1 + (1 ¡ n)x2 + ¢ ¢ ¢ + xn = 0

¢ ¢ ¢ = 0

x1 + x2 + ¢ ¢ ¢ + (1 ¡ n)xn = 0:

Find the reduced row{echelon form of A and hence, or otherwise, prove that

the solution of the above system is x1 = x2 = ¢ ¢ ¢ = xn, with xn arbitrary.

10. Let A =

·

a b

c d

¸

be a matrix over a ¯eld F. Prove that A is row{

equivalent to

·

1 0

0 1

¸

if ad ¡ bc 6= 0, but is row{equivalent to a matrix

whose second row is zero, if ad ¡ bc = 0.

20 CHAPTER 1. LINEAR EQUATIONS

11. For which rational numbers a does the following system have (i) no

solutions (ii) exactly one solution (iii) in¯nitely many solutions?

x + 2y ¡ 3z = 4

3x ¡ y + 5z = 2

4x + y + (a2 ¡ 14)z = a + 2:

[Answer: a = ¡4, no solution; a = 4, in¯nitely many solutions; a 6= §4,

exactly one solution.]

12. Solve the following system of homogeneous equations over Z2:

x1 + x3 + x5 = 0

x2 + x4 + x5 = 0

x1 + x2 + x3 + x4 = 0

x3 + x4 = 0:

[Answer: x1 = x2 = x4 + x5; x3 = x4, with x4 and x5 arbitrary elements of

Z2.]

13. Solve the following systems of linear equations over Z5:

(a) 2x + y + 3z = 4 (b) 2x + y + 3z = 4

4x + y + 4z = 1 4x + y + 4z = 1

3x + y + 2z = 0 x + y = 3:

[Answer: (a) x = 1; y = 2; z = 0; (b) x = 1 + 2z; y = 2 + 3z, with z an

arbitrary element of Z5.]

14. If (®1; : : : ; ®n) and (¯1; : : : ; ¯n) are solutions of a system of linear equa-

tions, prove that

((1 ¡ t)®1 + t¯1; : : : ; (1 ¡ t)®n + t¯n)

is also a solution.

15. If (®1; : : : ; ®n) is a solution of a system of linear equations, prove that

the complete solution is given by x1 = ®1 + y1; : : : ; xn = ®n + yn, where

(y1; : : : ; yn) is the general solution of the associated homogeneous system.

1.6. PROBLEMS 21

16. Find the values of a and b for which the following system is consistent.

Also ¯nd the complete solution when a = b = 2.

x + y ¡ z + w = 1

ax + y + z + w = b

3x + 2y + aw = 1 + a:

[Answer: a 6= 2 or a = 2 = b; x = 1 ¡ 2z; y = 3z ¡ w, with z; w arbitrary.]

17. Let F = f0; 1; a; bg be a ¯eld consisting of 4 elements.

(a) Determine the addition and multiplication tables of F. (Hint: Prove

that the elements 1+0; 1+1; 1+a; 1+b are distinct and deduce that

1 + 1 + 1 + 1 = 0; then deduce that 1 + 1 = 0.)

(b) A matrix A, whose elements belong to F, is de¯ned by

A =

2

4

1 a b a

a b b 1

1 1 1 a

3

5 ;

prove that the reduced row{echelon form of A is given by the matrix

B =

2

4

1 0 0 0

0 1 0 b

0 0 1 1

3

5 :

22 CHAPTER 1. LINEAR EQUATIONS

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