The left-hand member of this equation is what is called the determinant of the given system of equations.

We proceed to investigate the law of its formation. 3. First, suppose we have two equations,

Multiply the first by b,, the second by a,, and subtract, and we get

Hence

ab-ab1 = 0.

We may remark in passing that we shall obtain the same result by eliminating A, A, between the equations

A like theorem will be proved to be true for all determinants.

4. Next, suppose we have the three equations

Multiply these equations in order by the arbitrary multipliers A,,,,,, and add them together. Let the two ratios λ,λ,λ be determined by the conditions that the coefficients of x and x, in the resulting equation shall each be zero, i. e. let

The resulting equation is then reduced to

(a‚λ ̧ + b1λ ̧+c ̧λ ̧) x ̧ = 0,

which requires that

=

.....

Multiply the first of equations (A) by a,, the second by a,, and subtract, we then get

Hence, dividing each term of (B) by the corresponding member of (C) we get

It will be seen that the above process is really equivalent to that of eliminating A,, A,, A, between the equations (A) and (B). Hence

5. Next, let us have the four equations

а1x1+ α2x2+ α ̧¤ ̧+α ̧x ̧=0,

b1x1+b2x2+b2x3+b ̧ï ̧ = 0,

2

To effect the elimination, multiply the equations in order by λ, λ, λ, λ, add them, and equate the coefficients of x, x, x severally to zero. We shall then have

ɑ‚λ + bλ1⁄2 +c2λ ̧ + d2λ = 0]

аžλ1⁄2 + bgλ1⁄2 + €3λg + d2λ = 0

a ̧λ1+b ̧λ + çλ ̧+d‚ ̧λ = 0

which equations involve as a consequence

.(A'),

2

3

To determine the three ratios λ,: λ: λ: λ, multiply equations (A') in order by μ, M, M, add, and equate to zero the coefficients of A, A. We thus get

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Now, treating equations (C') as equations (A) were treated, we see that

These equations may be more conveniently written in the following equivalent forms:

27 39

4

Eliminating by means of these equations A,, λg, λg, λj from equation B', we get, as the result of the elimination of x1, x, x, x between the four given equations,

And since the above process is equivalent to the elimination of λ, λ, λ, λ between the equations (A') and (B′), we see that

6. The law of formation will be sufficiently obvious from the above investigations. If we have n lines and columns, it may be similarly proved that