# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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6. IfA = I3 2 -1

2 0 3

2 1

I -2 3

V 1 0

2 1

-2 3

1 0

-Ã

(c) A*

3 I, verify that 2(A + B) = 2A + 2B. 2

3 I ,and C = I 1 2 \0

1 0\

2 2 I, verify that

1 -1/

(a) (AB)C = A(BC)

(c) A(B + C) = AB + AC

(b) (A + B) + C = A + (B + C)

7. Prove each of the following laws of matrix algebra: (a) A + B = B + A (c) a(A + B) = aA + aB (e) A(BC) = (AB)C

(b) A + (B + C) = (A + B) + C (d) (a + â) A = aA + â A (f) A(B + C) = AB + AC

Ifx = I 32i

V - i>

(a) xTy (c) (x, y)

and y =

'-1 + i )

3i

, find

(b) yTy

(d) (y, y)

356

Chapter 7. Systems of First Order Linear Equations

/1 - H \

9. If x = I i I and y =

T T

(a) x y = y x

2\

3 - i I, show that ,1 + 2i

(b) (x, y) = (y,x)

In each of Problems 10 through 19 either compute the inverse of the given matrix, or else show that it is singular.

10.

12.

16.

18.

1 4

-2 3

1 2 1\

-2 1 8 )

1 -2 -7/

-1 -1\

2 1 0)

3 -2 1/

1 0 0 -1N

0 -1 1 0

-1 0 1 0

0 1 -1 1

11.

13.

15.

/2 17. I -1

19.

3 -1 \

6 2 1

1 1

2 -1 - 11

1 1 2

2 1 0 \

0 2 1

0 0 2 I

1

-1

1

-2

4 -1 -1

1

1

1)

-1

2 0\

-4 2

1 3

0 -1/

20. Prove that if there are two matrices B and C such that AB = I and AC = I, then B = C. This shows that a matrix A can have only one inverse.

3e2t \ e2t I, find

et 2e-t e2t\ ( 2et e-

2et e-t -e2t | and B(t) = |-et 2e-

et 3e-t 2e2t V 3et e-

(a) A + 3B (c) dA/dt

(b) AB

/

J0

(d) / A(t) dt

In each of Problems 22 through 24 verify that the given vector satisfies the given differential equation.

2) e2t

23. x' =

0 et+2«:

tet

24. x' =

Ï 1

21

01

1'

6t -8 \ e-t + 2

0

1 I e2t

x

x

x

7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 357

In each of Problems 25 and 26 verify that the given matrix satisfies the given differential equation.

bJ 1

.

=

4 -2/

/1 -1

26. = I3 2

2 1

H(t) =

e

-4e

( et e-2t e3^

H(t) = I -4e‘ -e-2t 2e3t

\ _et -e-2t

7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors

®In this section we review some results from linear algebra that are important for the solution of systems of linear differential equations. Some of these results are easily proved and others are not; since we are interested simply in summarizing some useful information in compact form, we give no indication of proofs in either case. All the results in this section depend on some basic facts about the solution of systems of linear algebraic equations.

Systems of Linear Algebraic Equations. A set of n simultaneous linear algebraic equations in n variables,

a11 X1 + a12 X2 + •" + a1nXn = b1,

. (1)

an1 X1 + an2 X2 + •" + annXn = bn,

can be written as

Ax = b, (2)

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