Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

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ANALYTIC GEOMETRY

Consider, next, a determinant of the third order, with two
rows identical. Expand the determinant by the minors of
the third, or odd, row. Each of these minors has its two rows
identical and is, therefore, zero, since the theorem has been
proved for two-rowed determinants. Consequently, the given
determinant is zero.
Similarly, having proved the theorem for three-rowed determinants, we can prove it for a four-rowed determinant.
For, we have but to expand the four-rowed determinant by the
minors of a row which is not one of the two identical rows.
This expansion will have the value zero, since each of the
minors in question is a three-rowed determinant with two
identical rows.
The process perpetuates itself. Hence the theorem is true
for a determinant of any order.
The method of proof used here is known as mathematical
induction. The fact that the theorem is true for a two-rowed
determinant leads up to its truth for a three-rowed determinant, etc.
COROLLARY. If the elements of two rows (or columns) of a
determinant are proportional, the determinant has the value
zero.
For, each element of one of the two rows (or columns) in
question is by hypothesis a multiple, m, of the corresponding
element of the other. Thus m can be taken out from the first
of the two rows (or columns) as a factor (Th. 3). The two
rows (or columns) are then identical, and Theorem 4 can be
applied.
THEOREM 5. If each element of a row (or column) is the sum
of two quantities, the determinant can be written as the sum of
two determinants.
Denote the determinant by A and the elements of the column (or row) in question by mn + mnl, m2 + mnz, *... Denote by
A the determinant obtained by replacing all the m"s in A by
zeros, and by A' the determinant obtained by replacing all the