expression for the determinant becomes Σ( — )^ka₁βa_2aa_3γ...a_nν, viz. a and β are transposed, and it is clear that the number of transpositions necessary to convert the permutation βαγ...ν of the second suffixes to the natural order is changed by unity. Hence the transposition of columns merely changes the sign of the determinant. Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant.

Theorem—Interchange of any two rows or of any two columns merely changes the sign of the determinant.

Corollary—If any two rows or any two columns of a determinant be identical the value of the determinant is zero.

Minors of a Determinant—From the value of Δ we may separate those members which contain a particular element a_ik as a factor, and write the portion a_ik A_ik; A_k, the cofactor of a_ik , is called a minor of order n−1 of the determinant.

Now a₁₁A₁₁=Σ ± = a₁₁a₂₂a₃₃...a_nn, wherein a₁₁ is not to be changed, but the second suffixes in the product a₂₂a₃₃...a_nn assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.

Hence a₁₁A₁₁ = a₁₁a₂₂a₃₃...a_nn, where the cofactor of a₁₁ is clearly the determinant obtained by erasing the first row and the first column.

Similarly A_ik , the cofactor of a_ik, is shown to be the product of (—)^i+k and the determinant obtained by erasing from Δ the i⁠^th row and k⁠^th column. No member of a determinant can involve more than one element from the first row. Hence we have the development

⁠Δ = a₁₁A₁₁ +a₁₂A₁₂ +a₁₃A₁₃+...+a_1nA_1n, proceeding according to the elements of the first row and the corresponding minors.

Similarly we have a development proceeding according to the elements contained in any row or in any column, viz.

⁠Δ=ai1Ai1 +ai2Ai2 +ai3Ai3+...+ainAin	${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}}$(A)
⁠Δ=a1kA1k +a2kA2k +a3kA3k+...+ankAnk

This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved.

Since the determinant

vanishes identically; we have by development according to the elements of the first row
⁠a₂₁A₁₁+a₂₂A₁₂+a₂₃A₁₃+...a_2nA_1n=0;
and, in general, since
⁠a_i1A_i1+a_i2A_i2+a_i3A_i3+...a_inA_in=Δ,
if we suppose the i^th and k^th rows identical
⁠a_k1A_i1+a_k2A_i2+a_k3A_i3+...a_knA_in=0 (k ≷ i);
and proceeding by columns instead of rows,
⁠a_1iA_1k+a_2iA_2k+a_3iA_3k+...a_niA_nk=0 (k ≷ i)
identical relations always satisfied by these minors.

If in the first relation of (A) we write a_is = b_is+c_is+d_is+... we find that Σa_isA_is = Σb_isA_is + Σc_isA_is + Σd_isA_is +... so that Δ breaks up into a sum of determinants, and we also obtain a theorem for the addition of determinants which have n – 1 rows in common. If we multiply the elements of the second row by an arbitrary magnitude λ, and add to the corresponding elements of the first row, Δ becomes Σa_1sA_1s + λΣa_2sA_1s = Δ, showing that the value of the determinant is unchanged. In general we can prove in the same way the—

Theorem.—The value of a determinant is unchanged if we add to the elements of any row or column the corresponding elements of the other rows or other columns respectively each multiplied by an arbitrary magnitude, such magnitude remaining constant in respect of the elements in a particular row or a particular column.

Observation.—Every factor common to all the elements of a row or of a column is obviously a factor of the determinant, and may be taken outside the determinant brackets.

⁠=(β − α)(γ − α)(β − γ).

The minor A_ik is ∂Δ/∂a_ik, and is itself a determinant of order n−1. We may therefore differentiate again in regard to any element a_rs where r≷i, s≷k; we will thus obtain a minor of A_ik, which is a minor also of Δ of order n−2. It will be

and will be obtained by erasing from the determinant A_ik the row and column containing the element a_rs; this was originally the r^⁠th row and the s^th column of Δ; the r^⁠th row of Δ is the r^⁠th or (r–1)^th row of A_ik according as r⁠≷⁠i and the s^th column of Δ is the s^th or (s−1)^th column of A_ik according as s⁠≷⁠k. Hence, if T_ri denote the number of transpositions necessary to bring the succession ri into ascending order of magnitude, the sign to be attached to the determinant arrived at by erasing the i⁠^th and r⁠^th rows and the k⁠^th and s⁠^th columns from Δ in order produce

Similarly proceeding to the minors of order n−3, we find that

is obtained from Δ by erasing the i^⁠th, r^⁠th, t^⁠th, rows, the k^⁠th, s^⁠th, u^⁠th columns, and multiplying the resulting determinant by −1 raised to the power T_tri +T_usk +i+k+r+s+t+u and the general law is clear.

Corresponding Minors.—In obtaining the minor

and we would have erased in an exactly similar manner had we been forming the determinant associated with

In the latter case the sign is determined by −1 raised to the same power as before, with the exception that T_uks, replaces T_usk; but if one of these numbers be even the other must be uneven; hence

Moreover

where the determinant factor is given by the four points in which the deleting lines intersect. This determinant and that associated with

Similarly p lines of deletion intersecting in p² points yield corresponding determinants of orders p and n−p respectively. Recalling the formula

⁠Δ=a₁₁A₁₁+a₁₂A₁₂+a₁₃A₁₃+...+a_1nA_1n,

it will be seen that a_1k and A_1k involve corresponding determinants. Since A_1k is a determinant we similarly obtain

and thence

and as before

an important expansion of Δ.

Similarly

and the general theorem is manifest, and yields a development in a sum of products of corresponding determinants. If the j^th column be identical with the i^th the determinant Δ vanishes identically; hence if j be not equal to i, k, or r,

Similarly, by putting one or more of the deleted rows or columns equal to rows or columns which are not deleted, we obtain, with Laplace, a number of identities between products of determinants of complementary orders.

Multiplication.—From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of Δ= (a₁₁, a₂₂, ... a_nn) and D = (b₁₁, b₂₂, b_nn ) may be written as a determinant of order 2n, viz.

Multiply the 1^st, 2^nd ... n^th rows by b₁₁, b₁₂, ... b_1n respectively, and