222 general form of an equation

elementary row operations on the two matrices

simultaneously. Reduce Ato an identity matrix.

The second matrix produced is the inverse

matrix to A.

As an example, we compute the inverse matrix to the

matrix of coefficients above:

Thus

Solving Systems of Linear Equations

via Inverse Matrices

In our example:

y+ 3z= 0

2x+ 4y– 2z= 18

x+ 5y+ 3z= 14

if xdenotes the column vector of the variables:

then the system can be compactly written:

Ax= b

Multiplying through by the inverse matrix A–1 yields:

again making the solution to the system apparent. Of

course, the work of computing the inverse matrix is

equivalent to the original process of Gaussian elimina-

tion. This approach, however, has the advantage that it

can be readily applied to a different set of constant val-

ues bwithout repeating the elimination process.

general form of an equation A formula that

describes the general relationship between variables

without specifying the constants involved is called the

general form of the equation. For example, the general

form of a

QUADRATIC

equation in variable xis ax2+ bx

+ c= 0. (The equation 2x2– 3x+ 4 = 0, for instance, is

a specific quadratic equation.) The formula for the

AREA

Aof a

CIRCLE

, A= πr2, where ris the radius of

the circle, is also an equation in general form.

general linear group (full linear group) The set of

all invertible n×nsquare matrices with real or com-

plex entries forms a

GROUP

under the operation of

MATRIX

multiplication. This group is called the nth-

order general linear group and is denoted GLn. It is

straightforward to see that the four group axioms do

indeed hold:

Closure. If Aand Bhave inverses, then so does their

product AB: (AB)–1 = B–1A–1.

Identity. The identity matrix Iis invertible and so

belongs to this set.