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Quantification of any entity or concept requires association to a numerical scale, so as to permit subsequent abstract reasoning and objective comparability; hence, every measurement carried out in the physicochemical world leads to a number, or scalar. Such numbers may be integer, rational (if expressible in the form p/q, where p and q denote integer numbers), or irrational (if not expressible in the previous form, and normally appearing as an infinite, nonrecurring decimal). If considered together, rational and irrational numbers account for the whole of real numbers – each one represented by a point in a straight line domain.

Departing from real numbers, related (yet more general) concepts have been invented; this includes notably the complex numbers, z – defined as an ordered pair of two real numbers, say, z ≡ a + ιb, where a and b denote real numbers and ι denotes , the imaginary unit. Therefore, z is represented by a point in a plane domain. In the complex number system, a general nth degree polynomial equation holds exactly (and always) n roots, not necessarily distinct though – as originally realized by Italian mathematicians Niccolò F. Tartaglia and Gerolamo Cardano in the sixteenth century; many concepts relevant for engineering purposes, originally conceived to utilize real numbers (as the only ones adhering to physical evidence), may be easily generalized via complex numbers.

The next stage of informational content is vectors – each defined by a triplet (a,b,c), where c also denotes a real number; each one is represented by a point in a volume domain and is often denoted via a bold, lowercase letter (e.g. v). Their usual graphical representation is a straight, arrowed segment linking the origin of a Cartesian system of coordinates to said point – where length (equal to , as per Pythagoras’ theorem), coupled with orientations (as per tan{b/a} and tan{c/}) fully define the said triplet. An alternative representation is as [a b c] or – also termed row vector or column vector, respectively; when three column vectors are assembled together, say, , , and , a matrix results, viz. , termed tensor – which may also be obtained by joining three row vectors, say, [a₁ a₂ a₃], [b₁ b₂ b₃], and [c₁ c₂ c₃]. The concept of matrix may be generalized so as to encompass other possibilities of combination of numbers besides a (3 × 3) layout; in fact, a rectangular (p × q) matrix of the form , or [a_i,j ; i = 1, 2,…, p; j = 1, 2, …, q] for short, may easily be devised.

Matrices are particularly useful in that they permit algebraic operations (and the like) be performed once on a set of numbers simultaneously – thus dramatically contributing to bookkeeping, besides their help to structure mathematical reasoning. In specific situations, it is useful to design higher order number structures, such as arrays of (or block) matrices; for instance, may be also represented as , provided that, say, A_1,1 ≡ [a₁], A_1,2 ≡ [a₂ a₃], , and represent, in turn, smaller matrices. An issue of compatibility arises in terms of the sizes of said blocks, though; for a starting (p × q) matrix A, only (p₁ × q₁) A_1,1, (p₁ × q₂) A_1,2, (p₂ × q₁) A_2,1, and (p₂ × q₂) A_2,2 matrices are allowed – obviously with p₁ + p₂ = p and q₁ + q₂ = q.

One of the most powerful applications of matrices is in solving sets of linear algebraic equations, say,

(1.1)

and

(1.2)

in its simplest version – where a_1,1, a_1,2, a_2,1, a_2,2, b₁, and b₂ denote real numbers, and x₁ and x₂ denote variables; if a_1,1 ≠ 0 and a_1,1 a_2,2 − a_1,2 a_2,1 ≠ 0, then one may start by isolating x₁ in Eq. (1.1) as

(1.3)

and then replace it in Eq. (1.2) to obtain

(1.4)

After factoring x₂ out, Eq. (1.4) becomes

(1.5)

so isolation of x₂ eventually gives

(1.6)

– which yields a solution only when a_1,1 a_2,2 − a_1,2 a_2,1 ≠ 0; insertion of Eq. (1.6) back in Eq. (1.3) yields

(1.7)

thus justifying why a solution for x₁ requires a_1,1 ≠ 0, besides a_1,1 a_2,2 − a_1,2 a_2,1 ≠ 0 (as enforced from the very beginning). Equation (1.6) may be rewritten as

(1.8)

– provided that one defines

(1.9)

complemented with

(1.10)

the left‐hand sides of Eqs. (1.9) and (1.10) are termed (second‐order) determinants. If both sides of Eq. (1.2) were multiplied by −a_1,2/a_2,2, one would get

(1.11)

– so ordered addition of Eqs. (1.1) and (1.11) produces simply

(1.12)

after having x₁ factored out; upon multiplication of both sides by a_2,2, Eq.,(1.12) becomes

(1.13)

with isolation of x₁ unfolding

(1.14)

– a result compatible with Eq. (1.7), once the two fractions are lumped, a_1,2 a_2,1 b₁ canceled out with its negative afterward, and a_1,1 finally dropped from numerator and denominator. Recalling Eq. (1.10), one may redo Eq. (1.14) to

(1.15)

as long as

(1.16)

is put forward; all forms conveyed by Eqs. (1.9), (1.10), and (1.16) do indeed share the form

(1.17)

irrespective of the values taken individually by α_1,1, α_1,2, α_2,1, and α_2,2. This is why the concept of determinant was devised – representing a scalar, bearing the unique property that its calculation resorts to subtraction of the product of elements in the secondary diagonal from the product of elements in the main diagonal of the accompanying (2 × 2) matrix. In the case of Eq. (1.10), the representation is selected because the underlying set of algebraic equations, see Eqs. (1.1) and (1.2), holds indeed as coefficient matrix. If a set of p algebraic linear equations in p unknowns is considered, viz.

(1.18)

then the concept of determinant can be extended in very much the same way to produce

(1.19)

however, the mode of calculation of higher order determinants is more complex – as it requires previous conversion to p!/2 second‐order determinants (as will be explained in due course), with calculation of each one to follow Eq. (1.17).