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

**Download**(direct link)

**:**

**195**> 196 197 198 199 200 201 .. 609 >> Next

\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

5

2

5

row.

1

0

0

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

5

2

1

(e) Obtain zeros in the off-diagonal positions in the third column by adding (-1) times the third row to the first row and adding (-1) 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 with the original matrix A.

1

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 (a11 = 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) =

(x1(t )\

U(tv

A(t) =

( an(t)

\an1(t)

J1n

(t)

(25)

ann(t))

respectively.

The matrix A(t) is said to be continuous at t = t0 or on an interval a < t < â 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

d

daj

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 = QfÜ ai/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 n2/2

2/

0

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

where C is a constant matrix; (28)

(29)

(30)

dt(CA)=C f ¦

d dA dB

— (A + B) = — + —; dt( ) dt dt

d dB dA

— (AB) = A— + — B.

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 0\ / 4 -2

1. If A = I 3 2 -1 I and B = 1-1 5

\—2 1 3 V 6 1

(a) 2A + B (c) AB

(b) A - 4B (d) BA

2. If A = Ö + 1. 1 + 2i) and B = ({ 3. I, find

\3 + 2i 2 - i / \2 -2i '

(a) A - 2 B (c) AB

(b) 3A + B (d) BA

I -2

3. If A = I 1

2 -1

- 3 I and B

J

-Ï

-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 + 3i

find

(b) A

3 2 -1\

5. If A = I2 -1 2 )

V 2

( 1 -2 0

**195**> 196 197 198 199 200 201 .. 609 >> Next