add to the (n+1)^th row; by b₂₁, b₂₂ ... b_2ⁿ, and add to the (n+2)^th row; by b₃₁, b₃₂ ... b_3ⁿ and add to the (n+3)^rd row, &c. C then becomes

and all the elements of D become zero. Now by the expansion theorem the determinant becomes

⁠(−)^1+2+3+..+2n⁠B.C = (−1)^n(2n+1)+n⁠C⁠=⁠C.

We thus obtain for the product a determinant of order n. We may say that, in the resulting determinant, the element in the i⁠^th row and k^th column is obtained by multiplying the elements in the k^th row of the first determinant severally by the elements in the i⁠^th row of the second, and has the expression

⁠a_k1b_i1+a_k2b_i2+a_k3b_i3... +a_kn⁠b_in,

and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows.

Remark.—In particular the square of a determinant is a determinant of the same order (b₁₁b₂₂b₃₃ ...b_nn) such that b_ik = b_ki; it is for this reason termed symmetrical.

The Adjoint or Reciprocal Determinant arises from Δ = (a₁₁a₂₂a₃₃ ...a_nn) by substituting for each element A_ik the corresponding minor A_ik so as to form D = (A₁₁A₂₂A₃₃ A_nn). If we form the product Δ.D by the theorem for the multiplication of determinants we find that the element in the i⁠^th row and k^th column of the product is

⁠a_ki⁠A_i1+a_k2A_i2+...+a_kn⁠A_in,

the value of which is zero when k is different from i, whilst it has the value Δ when k=i. Hence the product determinant has the principal diagonal elements each equal to Δ and the remaining elements zero. Its value is therefore Δⁿ and we have the identity
⁠D.Δ = Δⁿ or D=Δ^n⁠–1.
It can now be proved that the first minor of the adjoint determinant, say B^rs is equal to Δ^n–2a_rs.

From the equations

⁠a₁₁x₁+ a₁₂x₂+ a₁₃x₃ +... = ξ₁ ,
⁠a₂₁x₁+ a₂₂x₂+ a₂₃x₃ +... = ξ₂ ,
⁠a₃₁x₁+ a₃₂x₂+ a₃₃x₃ +... = ξ₃ ,
we derive⁠.⁠.⁠.⁠...⁠.
⁠Δx₁ = A₁₁ξ₁+A₂₁ξ₂+A₃₁ξ₃+... ,
⁠Δx₂ = A₁₂ξ₁+A₂₂ξ₂+A₃₂ξ₃+... ,
⁠Δx₃ = A₁₃ξ₁+A₂₃ξ₂+A₃₃ξ₃+... ,
and thence⁠.⁠.⁠.⁠...⁠.
⁠Δ^n⁠–1ξ₁⁠=⁠B₁₁Δx₁+B₁₂Δx₂+B₁₃Δx₃+... ,
⁠Δ^n⁠–1ξ₂⁠=⁠B₂₁Δx₁+B₂₂Δx₂+B₂₃Δx₃+... ,
⁠Δ^n⁠–1ξ₃⁠=⁠B₃₁Δx₁+B₃₂Δx₂+B₃₃Δx₃+... ,
⁠.⁠.⁠.⁠...⁠.
and comparison of the first and third systems yields

⁠B_rs = Δ^n⁠–2a_rs.

In general it can be proved that any minor of order p of the adjoint is equal to the complementary of the corresponding minor of the original multiplied by the (p – 1)^th power of the original determinant.

Theorem.—The adjoint determinant is the (n – 1)^th power of the original determinant. The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

Determinants of Special Forms.—It was observed above that the square of a determinant when expressed as a determinant of the same order is such that its elements have the property expressed by a_ik = a_ki. Such determinants are called symmetrical. It is easy to see that the adjoint determinant is also symmetrical, viz. such that A_ik = A_ki, for the determinant got by suppressing the i⁠^th row and k^th column differs only by an interchange of rows and columns from that got by suppressing the k^th row and i⁠^th column. If any symmetrical determinant vanish and be bordered as shown below

it is a perfect square when considered as a function of λ₁, λ₂, λ₃. For since A₁₁ A₂₂ −A_2⁠
12 = Δa₃₃, with similar relations, we have a number of relations similar to A₁₁A₁₂=A_2 
12, and either A_rs = +√ (A_rrA_ss) or − √ (A_rrA_ss) for all different values of r and s. Now the determinant has the value

⁠–λ_2
1A₁₁ + λ_2
2A₂₂ + λ_2
3A₃₃ + 2λ₂λ₃A₂₃ + 2λ₃λ₁A₃₁ + 2λ₁λ₂A₁₂}
⁠ = –Σλ_2
rA_rr – 2Σλ_rλ_sA_rs in general, and hence by substitution
⁠±λ₁√A₁₁ + λ₂√A₂₂ + ... λ_n √ A_nn}².

A skew symmetric determinant has a_rr = 0 and a_rs = – a_sr for all values of r and s. Such a determinant when of uneven degree vanishes, for if we multiply each row by –1 we multiply the determinant by (–1)ⁿ = –1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered. Hence Δ = –Δ or Δ = 0. When a skew symmetric determinant is of even degree it is a perfect square. This theorem is due to Cayley, and reference may be made to Salmon’s Higher Algebra, 4th ed. Art. 39. In the case of the determinant of order 4 the square root is

⁠A₁₂A₃₄ – A₁₃ A₂₄ + A₁₄A₂₃.

A skew determinant is one which is skew symmetric in all respects, except that the elements of the leading diagonal are not all zero. Such a determinant is of importance in the theory of orthogonal substitution. In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions
⁠X =⁠ax +⁠by +⁠cz,
⁠Y = a′x +⁠b′y +⁠c′z,
⁠Z = a″x + b″y + c″z,
where X² + Y² + Z² = x² + y² + z². This relation implies six equations. between the coefficients, so that only three of them are independent. Further we find

⁠x = aX + a′Y + a″Z,
⁠y = bX + b′Y + b″Z,
⁠z = cX + c′Y + c″Z,
and the problem is to express the nine coefficients in terms of three independent quantities.

In general in space of n dimensions we have n substitutions similar to

⁠X₁ = a₁₁x₁ +a₁₂x₂ + ... + a_1n⁠x_n,

and we have to express the n² coefficients in terms of ½n(n – 1) independent quantities; which must be possible, because

⁠X_2
1+X_2
2+ ... + X_2
n =x_2
1 + x_2
2 + x_2
3 + ... + x_2
n .

Let there be 2n equations

⁠x₁= b₁₁ξ₁ + b₁₂ξ₂ + b₁₃ξ₃ + ...,
⁠x₁= b₂₁ξ₁ + b₂₂ξ₂ + b₂₃ξ₃ + ...,
⁠.⁠.⁠.⁠.⁠.
⁠X₁= b₁₁ξ₁ + b₂₁ξ₂ + b₃₁ξ₃ + ...,
⁠X₂= b₁₂ξ₁ + b₂₂ξ₂ + b₃₂ξ₃ + ...,
⁠.⁠.⁠.⁠.⁠.

where b_rr = 1 and b_rs = – b_sr for all values of r and s. There are then ½n(n–1) quantities b_rs . Let the determinant of the b’s be Δ_b and B_rs, the minor corresponding to b_rs . We can eliminate the quantities ξ₁, ξ₂, ... ξ_n and obtain n relations

⁠Δ_bX₁ = (2B₁₁ – Δ_b)x₁⁠ +2B₂₁x₂+2B₃₁x₃+...,
⁠Δ_bX₂ = ⁠2B₁₂x₁+ (2B₂₂ – Δ_b)x₂ + 2B₃₂x₃+...,
⁠.⁠.⁠.⁠.⁠.⁠.⁠.⁠.

and from these another equivalent set

⁠Δ_bx₁ = (2B₁₁ – Δ_b)X₁⁠ +2B₁₂X₂+2B₁₃X₃+...,
⁠Δ_bx₂ = ⁠2B₁₂X₁+ (2B₂₂ – Δ_b)X₂ + 2B₂₃x₃+...,
⁠.⁠.⁠.⁠.⁠.⁠.⁠.⁠.

and now writing

⁠2B_ii – Δ_b/Δ_b=a_ii,2B_ik/Δ_b = a_ik,

we have a transformation which is orthogonal, because ΣX² = Σx² and the elements a_ii, a_ik are functions of the ½n(n − 1) independent quantities b. We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the ½n(n − 1) quantities b are clearly arbitrary.

For the second order we may take

and the adjoint determinant is the same; hence

⁠(1 + λ²)x₁ = (1-λ²)X₁ +⁠2λX₂,
⁠(1 + λ²)x₂⁠= –2λX₁ + (1 – λ²)X₂,

Similarly, for the order 3, we take

and the adjoint is

leading to the orthogonal substitution

⁠Δ_bx₁ = (1 +λ² − μ² − ν²)X₁⁠+2(ν + λμ)X₂⁠+2(− μ + λν)X₃
⁠Δ_bx₂ = ⁠2(λμ − ν)X₁ + (1 + μ² − λ² − ν²)X₂⁠+2(μν + λ)X₃
⁠Δ_bx₃ = ⁠2(λν + μ)X₁⁠+2(μν − λ)X₂+(1 +ν² − λ² − μ²)X₃.

Functional determinants were first investigated by Jacobi in a work De Determinantibus Functionalibus. Suppose n dependent variables y₁, y₂,...y_n, each of which is a function of n independent variables x₁, x₂,...x_n, so that y_s= ƒ_s (x₁, x₂,...x_n). From the differential coefficients of the y’s with regard to the x’s we form the functional determinant