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Quiz 19

1. If 0 is an eigenvalue of the matrix of coeffi-

cients of a homogeneous system of n linear

equations in n unknowns, then the system

has infinitely many solutions.

(a) Always true.

(b) Sometimes true.

(c) Never true.

(d) None of the above.

1

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2. If the reduced row echelon form of the matrix

of coefficients of a system of n linear equa-

tions in n unknowns is In, then the matrix of

coefficients is singular.

(a) Always true.

(b) Sometimes true.

(c) Never true.

(d) None of the above.

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3. If the rank of the augmented matrix of a sys-

tem of n linear equations in n unknowns is

greater than the rank of the matrix of coef-

ficients, then the system is inconsistent.

(a) Always true.

(b) Sometimes true.

(c) Never true.

(d) None of the above.

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4. If 0 is an eigenvalue of the matrix of coef-

ficients of a system of n linear equations in

n unknowns, then the system has infinitely

many solutions.

(a) Always true.

(b) Sometimes true.

(c) Never true.

(d) None of the above.

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5. The general solution of x =

0 16 5

x is:

(a) x= C1e3t 3

1

+ C2e2t 2

1

(b) x= C1e3t

10

+ C2e

2t

11

(c) x= C1e3t 1

3

+ C2e2t 1

2

(d) x= C1e3t

12

+ C2e

2t

13

(e) None of the above.

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6. The general solution of x =

0 1 00 0 1

1 2 8 1

x

is: (Hint: 3 is an eigenvalue.) x=

(a) C1e3t

13

9

+C2e2t

12

4

+C3e2t

12

4

(b) C1e3t 1

39

+C2e2t 1

24

+C3te2t 1

24

(c) C1e3t

13

9

+ C2e2t

12

4

+

C3

e2t 104

+ te2t 124

(d) C1e3t

1

39

+ C2e

2t

124

+

C3

e2t

10

4

+ te2t

12

4

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7. The solution of the initial-value problem

x =

1 1 20 2 2

1 1 3

x; x(0) =

20

1

is: HINT: 2 is a root of the characteristic

polynomial.

(a) x= 2e2t

11

0

et

02

1

(b) x= e3t

22

1

e2t

11

0

+ et

02

1

(c) x= 2e3t 22

1

3e2t 110

+ et 021

(d) x= 3e2t

110

e3t

221

(e) None of the above.

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8. The general solution of

x =

2 2 10 1 0

2 2 1

x

is: (Hint: 3 is an eigenvalue.)

(a) x= C1e3t

10

1

+C2e

t

111

+C3e

t

201

(b) x= C1e3t

10

1

+C2et

11

1

+C3

20

1

(c) x= C1e3t 1

01

+C2e

t 111

+C3et 1

02

(d) x= C1e3t

10

1

+C2et

11

1

+C3

10

2

(e) None of the above.

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9. The general solution of

x =

2 2 62 1 3

2 1 1

x

is: (Hint: 6 is an eigenvalue.)

x=

(a) C1e6t

2

11

+ C2e

2t3

02

+ C3

1

20

(b) C1e6t

21

1

+ C2e2t

30

2

(c) C1e6t

21

1

+C2e2t

30

2

+C3e2t

12

0

(d) C1e6t

21

1

+ C2

30

2

+ C3e2t

12

0

(e) None of the above.

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10. A fundamental set of solutions of

x =

1 21 3

x

is:

(a)

e2t

cos t

11

sin t

1

0

,

e2t

cos t

1

0

+ sin t

11

(b)

e2t

cos t

11

sin t

1

0

,

e2t

cos t

1

0

+ sin t

11

(c)

e2t

cos t

11

+ sin t

1

0

,

e2t

cos t

1

0

+ sin t

11

(d)

e

2t cos t

11

+ sin t 1

0

,

e2t

cos t

1

0

sin t

11

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11. The general solution of x =

4 15 2

x is:

(a) x= C1e3t

cos 2t 1

1

sin 2t 0

2

+

C2e3t

cos 2t

02

+ sin 2t

11

(b) x = C1e3t

cos 2t

11

sin 2t

02

+

C2e3t

cos 2t

02

+ sin 2t

11

(c)

x=C1e3t

cos 2t

11

sin 2t

02

+

C2e3t

cos 2t 0

2

+ sin 2t 1

1

(d) x = C1e2t

cos 3t

11

sin 3t

02

+

C2e2t cos 3t

02 + sin 3t

11

(e) None of the above.

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13. A fundamental set of solutions of

x =

2 1 13 3 43 1 2

xis:

(a) x1 = et

01

1

, x2=e2t

11

1

(b) x1 = et

01

1

, x2 = e2t

11

1

, x3 =

te2t 1

11

(c) x1 = et

01

1

, x2 = e2t

11

1

, x3 =

e2t 01

0

+ te2t 111

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(d) x1 = et

01

1

, x2 = e2t

11

1

, x3 =

e2t 11

1

+ te2t0

10

(e) None of the above.