as to each column), and such that only the sign is
altered when any two columns are interchanged; these properties
completely determine the function, except as to a common factor which
may multiply all the terms. If, to get rid of this arbitrary common
factor, we assume that the product of the elements in the dexter
diagonal has the coefficient +1, we have a complete definition of the
determinant, and it is interesting to show how from these properties,
assumed for the definition of the determinant, it at once appears that
the determinant is a function serving for the solution of a system of
linear equations. Observe that the properties show at once that if any
column is = 0 (that is, if the elements in the column are each = 0),
then the determinant is = 0; and further, that if any two columns are
identical, then the determinant is = 0.

5. Reverting to the system of linear equations written down at the
beginning of this article, consider the determinant

|ax + by + cz - d , b , c |;
|a'x + b'y + c'z - d', b', c'|
|a"x + b"y + c"z - d", b", c"|

it appears that this is

= x|a , b , c | + y|b , b , c | + z|c , b , c | - |d , b , c |;
|a', b', c'| |b', b', c'| |c', b', c'| |d', b', c'|
|a", b", c"| |b", b", c"| |c", b", c"| |d", b", c"|

viz. the second and third terms each vanishing, it is

= x|a , b , c | - |d , b , c |.
|a', b', c'| |d', b', c'|
|a", b", c"| |d", b", c"|

But if the linear equations hold good, then the first column of the
original determinant is = 0, and therefore the determinant itself is = 0;
that is, the linear equations give

x|a , b , c | - |d , b , c | = 0;
|a', b', c'| |d', b', c'|
|a", b", c"| |d", b", c"|

which is the result obtained above.

We might in a similar way find the values of y and z, but there is a
more symmetrical process. Join to the original equations the new
equation

[alpha]x + [beta]y + [gamma]z = [delta];

a like process shows that, the equations being satisfied, we have

|[alpha], [beta], [gamma], [delta]| = 0;
| a , b , c , d |
| a' , b' , c' , d' |
| a" , b" , c" , d" |

or, as this may be written,

|[alpha], [beta], [gamma] | - [delta]| a , b , c | =