282 inverse of a statement

π

–

2

π

–

2

π

–

2

π

–

2

dy

––

dx

with determinant det(A) = ad – bc is the matrix:

The process of G

AUSSIAN ELIMINATION

provides a rela-

tively straightforward method for computing the

matrix inverse of any square matrix of a larger size.

If, in a system of n

SIMULTANEOUS LINEAR EQUA

-

TIONS

Ax= b, the matrix Aof coefficients is invertible,

then the system has solution given by x= A–1b. In par-

ticular, if ejdenotes the column vector whose only

nonzero entry is a 1 in the jth position, then xj= A–1ej

is the jth column of A–1. That is, the jth column of this

inverse matrix is a solution to the system of equations

Axj= ej. By C

RAMER

’

S RULE

, the ith entry of this col-

umn, that is the (i,j)th entry of the inverse matrix, is the

ratio of determinants:

where A|ij is the matrix Awith the ith column replaced

by ej. Computing det(A|ij) is equivalent, up to a plus or

minus sign, to computing the determinant of the matrix

obtained from Aby deleting its ith column and jth row.

This value is sometimes called the (i,j)th cofactor of A.

If Ais invertible, then it is impossible to find a

nonzero column vector xsuch that Ax= 0. (Otherwise,

x= A–1 0 = 0.) This observation is important for the

study of

EIGENVECTOR

s and

EIGENVALUE

s.

See also

GENERAL LINEAR GROUP

.

inverse of a statement See

CONTRAPOSITIVE

.

inverse square law Any relationship between two

physical variables for which one is proportional to the

RECIPROCAL

of the square of the other is referred to as

an inverse square law. For example, the law of gravita-

tion as developed by S

IR

I

SAAC

N

EWTON

(1642–1727)

asserts that the magnitude Fof the gravitational force

between two bodies of masses mand Mis given by:

Here Gis the gravitational constant (equal to 6.67 ×

10–11 m3kg–1sec–2) and ris the distance between the

two masses. This is an inverse square law. The illumi-

nation provided by a source of light decreases by the

inverse of the square of the distance from the source

and so too is an inverse square relationship.

inverse trigonometric functions An

INVERSE FUNC

-

TION

to any trigonometric function is called an inverse

trigonometric function. For instance, the inverse sine of

a number x, written arcsinxor sin–1x, is an

ANGLE

afor

whose sine is x: sina= x. Since the sine curve adopts val-

ues only between –1 and 1, the inverse sine function is

defined only for values –1 ≤x≤1. One should also note

that for any value xthere are infinitely many angles a

with sin a= x. It is usually assumed then that the

angle ais chosen so that – ≤a≤. (This is called the

range of principal values for sine.) Similarly the inverse

cosine of a number xwith –1 ≤x≤1, written arccos x

or cos–1 x, is an angle a, usually chosen in the principal

range for cosine, 0 ≤a≤π, with cos a= x. Since the tan-

gent function adopts all real values, the inverse tangent

function is defined for any real number x, and arctan x,

or tan–1 x, is defined to be that angle ain the principal

range for tangent, – ≤a≤, such that tan a= x.

The inverse trigonometric functions have the fol-

lowing

DERIVATIVE

s:

These can be established by making use of the relation

sin2y+ cos2y= 1. For instance, to compute the deriva-

tive of y= sin–1 x, write sin y= xand then differentiate

this equation making use of the

CHAIN RULE

. This

yields cos y = 1, thereby establishing:

as claimed. The T

AYLOR SERIES

of the arctan function

gives G

REGORY

’

S SERIES

.