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2.7 PROBLEMS

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1. Let A =

1 4

¡3 1

. Prove that A is non{singular, ¯nd A¡1 and

express A as a product of elementary row matrices.

[Answer: A¡1 =

· 1

13 ¡ 4

A = E21(¡3)E2(13)E12(4) is one such decomposition.]

2. A square matrix D = [dij ] is called diagonal if dij = 0 for i 6= j. (That

is the o®{diagonal elements are zero.) Prove that pre{multiplication

of a matrix A by a diagonal matrix D results in matrix DA whose

rows are the rows of A multiplied by the respective diagonal elements

of D. State and prove a similar result for post{multiplication by a

diagonal matrix.

Let diag (a1; : : : ; an) denote the diagonal matrix whose diagonal ele-

ments dii are a1; : : : ; an, respectively. Show that

diag (a1; : : : ; an)diag (b1; : : : ; bn) = diag (a1b1; : : : ; anbn)

and deduce that if a1 : : : an 6= 0, then diag (a1; : : : ; an) is non{singular

and

(diag (a1; : : : ; an))¡1 = diag (a¡1

1 ; : : : ; a¡1

n ):

Also prove that diag (a1; : : : ; an) is singular if ai = 0 for some i.

3. Let A =

0 0 2

1 2 6

3 7 9

5. Prove that A is non{singular, ¯nd A¡1 and

express A as a product of elementary row matrices.

[Answers: A¡1 =

4 ¡12 7 ¡2

2 ¡3 1

2 0 0

A = E12E31(3)E23E3(2)E12(2)E13(24)E23(¡9) is one such decompo-

sition.]

50 CHAPTER 2. MATRICES

4. Find the rational number k for which the matrix A =

1 2 k

3 ¡1 1

5 3 ¡5

is singular. [Answer: k = ¡3.]

5. Prove that A =

1 2

¡2 ¡4

is singular and ¯nd a non{singular matrix

P such that PA has last row zero.

6. If A =

1 4

¡3 1

, verify that A2 ¡ 2A + 13I2 = 0 and deduce that

A¡1 = ¡ 1

13 (A ¡ 2I2).

7. Let A =

1 1 ¡1

0 0 1

2 1 2

(i) Verify that A3 = 3A2 ¡ 3A + I3.

(ii) Express A4 in terms of A2; A and I3 and hence calculate A4

explicitly.

(iii) Use (i) to prove that A is non{singular and ¯nd A¡1 explicitly.

[Answers: (ii) A4 = 6A2 ¡ 8A + 3I3 =

4 ¡11 ¡8 ¡4

12 9 4

20 16 5

(iii) A¡1 = A2 ¡ 3A + 3I3 =

4 ¡1 ¡3 1

2 4 ¡1

0 1 0

5.]

8. (i) Let B be an n£n matrix such that B3 = 0. If A = In¡B, prove

that A is non{singular and A¡1 = In + B + B2.

Show that the system of linear equations AX = b has the solution

X = b + Bb + B2b:

(ii) If B =

0 r s

0 0 t

0 0 0

5, verify that B3 = 0 and use (i) to determine

(I3 ¡ B)¡1 explicitly.

2.7. PROBLEMS 51

[Answer:

1 r s + rt

0 1 t

0 0 1

5.]

9. Let A be n £ n.

(i) If A2 = 0, prove that A is singular.

(ii) If A2 = A and A 6= In, prove that A is singular.

10. Use Question 7 to solve the system of equations

x + y ¡ z = a

z = b

2x + y + 2z = c

where a; b; c are given rationals. Check your answer using the Gauss{

Jordan algorithm.

[Answer: x = ¡a ¡ 3b + c; y = 2a + 4b ¡ c; z = b.]

11. Determine explicitly the following products of 3 £ 3 elementary row

matrices.

(i) E12E23 (ii) E1(5)E12 (iii) E12(3)E21(¡3) (iv) (E1(100))¡1

(v) E¡1

12 (vi) (E12(7))¡1 (vii) (E12(7)E31(1))¡1.

[Answers: (i)

0 0 1

1 0 0

0 1 0

5 (ii)

0 5 0

1 0 0

0 0 1

5 (iii)

4 ¡8 3 0

¡3 1 0

0 0 1

(iv)

100 0 0

0 1 0

0 0 1

5 (v)

0 1 0

1 0 0

0 0 1

5 (vi)

1 ¡7 0

0 1 0

0 0 1

5 (vii)

1 ¡7 0

0 1 0

¡1 7 1

5.]

12. Let A be the following product of 4 £ 4 elementary row matrices:

A = E3(2)E14E42(3):

Find A and A¡1 explicitly.

[Answers: A =

664

0 3 0 1

0 1 0 0

0 0 2 0

1 0 0 0

775

; A¡1 =

664

0 0 0 1

0 1 0 0

0 0 1

2 0

1 ¡3 0 0

775

52 CHAPTER 2. MATRICES

13. Determine which of the following matrices over Z2 are non{singular

and ¯nd the inverse, where possible.

(a)

664

1 1 0 1

0 0 1 1

1 1 1 1

1 0 0 1

775

(b)

664

1 1 0 1

0 1 1 1

1 0 1 0

1 1 0 1

775

[Answer: (a)

664

1 1 1 1

1 0 0 1

1 0 1 0

1 1 1 0

775

14. Determine which of the following matrices are non{singular and ¯nd

the inverse, where possible.

(a)

1 1 1

¡1 1 0

2 0 0

5 (b)

2 2 4

1 0 1

0 1 0

5 (c)

4 6 ¡3

0 0 7

0 0 5

(d)

2 0 0

0 ¡5 0

0 0 7

5 (e)

664

1 2 4 6

0 1 2 0

0 0 1 2

0 0 0 2

775

(f)

1 2 3

4 5 6

5 7 9

[Answers: (a)

0 0 1

0 1 1

1 ¡1 ¡1

5 (b)

4 ¡1

2 2 1

0 0 1

2 ¡1 ¡1

5 (d)

2 0 0

0 ¡1

5 0

0 0 1

(e)

664

1 ¡2 0 ¡3

0 1 ¡2 2

0 0 1 ¡1

0 0 0 1

775

15. Let A be a non{singular n £ n matrix. Prove that At is non{singular

and that (At)¡1 = (A¡1)t.

16. Prove that A =

a b

c d

has no inverse if ad ¡ bc = 0.

[Hint: Use the equation A2 ¡ (a + d)A + (ad ¡ bc)I2 = 0.]

2.7. PROBLEMS 53

17. Prove that the real matrix A =

1 a b

¡a 1 c

¡b ¡c 1

5 is non{singular by

proving that A is row{equivalent to I3.

18. If P¡1AP = B, prove that P¡1AnP = Bn for n ¸ 1.

19. Let A =

· 2

; P =

1 3

¡1 4

. Verify that P¡1AP =

· 5

12 0

0 1

and deduce that

An =

3 3

4 4

¶n ·

4 ¡3

¡4 3

20. Let A =

a b

c d

be a Markov matrix; that is a matrix whose elements

are non{negative and satisfy a+c = 1 = b+d. Also let P =

b 1

c ¡1

Prove that if A 6= I2 then

(i) P is non{singular and P¡1AP =

1 0

0 a + d ¡ 1

(ii) An !

b + c

b b

c c

as n ! 1, if A 6=

0 1

1 0

21. If X =

1 2

3 4

5 6

5 and Y =

4 ¡1

5, ¯nd XXt; XtX; Y Y t; Y tY .

[Answers:

5 11 17

11 25 39

17 39 61

5 ;

35 44

44 56

;

1 ¡3 ¡4

¡3 9 12

¡4 12 16

5 ; 26.]

22. Prove that the system of linear equations

x + 2y = 4

x + y = 5

3x + 5y = 12

is inconsistent and ¯nd a least squares solution of the system.

[Answer: x = 6; y = ¡7=6.]

54 CHAPTER 2. MATRICES

23. The points (0; 0); (1; 0); (2; ¡1); (3; 4); (4; 8) are required to lie on a

parabola y = a + bx + cx2. Find a least squares solution for a; b; c.

Also prove that no parabola passes through these points.

[Answer: a = 1

5 ; b = ¡2; c = 1.]

24. If A is a symmetric n£n real matrix and B is n£m, prove that BtAB

is a symmetric m £ m matrix.

25. If A is m £ n and B is n £ m, prove that AB is singular if m > n.

26. Let A and B be n £ n. If A or B is singular, prove that AB is also

singular.

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