# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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3For large n the number of multiplications required to evaluate A—1 by Eq. (24) is proportional to n!. If one uses more efficient methods, such as the row reduction procedure described later, the number of multiplications is proportional only to n3. Even for small values of n (such as n = 4), determinants are not an economical tool in calculating inverses, and row reduction methods are preferred.

7.2 Review of Matrices

353

EXAMPLE

2

Find the inverse of

/1 -1 -1N

A = I 3 -1 2

V2 2 3

The matrix A can be transformed into I by the following sequence of operations. The result of each step appears below the statement.

(a) Obtain zeros in the off-diagonal positions in the first column by adding (-3) times the first row to the second row and adding (-2) times the first row to the third row.

'1 -1 -1N

0 2 5

,0 4 5,

(b)

Obtain a 1 in the diagonal position in the second column by multiplying the second row by 2.

-1\

(c)

/1 -1 0 1

\0 4

Obtain zeros in the off-diagonal positions in the second column by adding the second row to the first row and adding (—4) times the second row to the third

row.

/1

0

Vo

2^

5

2

-5/

(d)

Obtain a 1 in the diagonal position in the third column by multiplying the third

row by (-5).

1

0

0

2^

5

2

1

(e) Obtain zeros in the off-diagonal positions in the third column by adding (-2) times the third row to the first row and adding (- 2) times the third row to the second row.

If we perform the same sequence of operations in the same order on I, we obtain the following sequence of matrices:

( 1 0 0\

V-2 0 1/

0

354

Chapter 7. Systems of First Order Linear Equations

0\

0

/-

V 4 -2 \)

The last of these matrices is A 1, with the original matrix A.

0

0

5/

( - -- -\

10 10 10

1 _ 1 1

2 2 2

\-4 2 1

V 5 5

a result that can be verified by direct multiplication

3

2

2

2

This example is made slightly simpler by the fact that the original matrix A had a 1 in the upper left corner (ajj = 1). If this is not the case, then the first step is to produce a 1 there by multiplying the first row by 1/an, as long as a11 = 0. If a11 = 0, then the first row must be interchanged with some other row to bring a nonzero element into the upper left position before proceeding.

Matrix Functions. We sometimes need to consider vectors or matrices whose elements are functions of a real variable t. We write

x(t ) =

A(t ) =

( an(t)

\3n1(t )

(t)

(25)

ann(t ))

respectively.

The matrix A(t) is said to be continuous at t = t0 or on an interval a < t < 3 if each element of A is a continuous function at the given point or on the given interval. Similarly, A(t) is said to be differentiable if each of its elements is differentiable, and its derivative dA/dt is defined by

dA

~dt

dBj

dt

(26)

that is, each element of dA/dt is the derivative of the corresponding element of A. In the same way the integral of a matrix function is defined as

j* A(t) dt = Qf6 atJ(t) dt

(27)

For example, if

then

A'(0 =

cos t 0

A(t ) =

1

sint

sin t t

1 cos t

J* A(t) dt = ?2 -2/2

Many of the rules of elementary calculus extend easily to matrix functions; in particular,

where C is a constant matrix; (28)

(29)

(30)

d dA

— (CA) = C—,

d^^ dt

d dA dB

— (A + B) =-------------1-----;

dt( ) dt dt

d (AB) = A ™ + dAB.

dt( ) dt dt

7.2 Review of Matrices

355

PROBLEMS

In Eqs. (28) and (30) care must be taken in each term to avoid interchanging the order of multiplication. The definitions expressed by Eqs. (26) and (27) also apply as special cases to vectors.

( 1 -2

1. If A = I 3 2

\—2 1

0\ /4 -2

— 1 I and B = I — 1 5

(a) 2A + B (c) AB

(b) A - 4B (d) BA

2. If A = (1 + i. - + 2i| and B:

\3 + 2i 2 - i

1 3

2 2i

find

(a) A - 2 B (c) AB

(b) 3A + B (d) BA

(-2

3. If A = I 1

21

- 3 I and B

J

-1 I, find

0

(a) AT (c) AT + BT

(b) BT (d) (A + B)T

4. If A =

(a) AT

3 2i

1 + i

2 - i -2 + 31-

find

(b) A

3 2 -1

5. If A = (2 -1 2|

1 2 1

/ 1 -2 0

6. IfA = ( 3 2 -1

2 0 3

-1 I, B= I- 2

2 1

(-2 3

1 0

2 1

-2 3

1 0

1

(c) A*

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

3 I, and C = I 1 2 \0

1 °\

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) = a A + aB (e) A(BC) = (AB)C

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

If x =

2

3i

1i

and y =

-1 + iI

3i

, find

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

(b) yTy

(d) (y, y)

356

Chapter 7. Systems of First Order Linear Equations

/1 - H \ / 2 \

9. If x = I i \ and y = I 3 — i \, show that

12 ; y u u)

T T

(a) x y = y x

(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 23

1 2 1

-2 1 8 )

1 -2 -V

-1 -1\

12 1 o\

3 -2 1

1 0 0 -1N

0 -1 1 0

-1 0 1 0

0 1 -1 1

11.

13.

15.

19.

3 -1 \

6 2 )

1 1

2 -1 - 11

1 1 2

2 1 0 \

0 2 1

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