F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

and

(1.2)

in its simplest version – where a 1,1, a 1,2, a 2,1, a 2,2, b 1, and b 2denote real numbers, and x 1and x 2denote 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 x 1in Eq. (1.1)as

(1.3)

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

(1.4)

After factoring x 2out, Eq. (1.4)becomes

(1.5)

so isolation of x 2eventually 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 x 1requires 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 x 1factored out; upon multiplication of both sides by a 2,2, Eq.,(1.12)becomes

(1.13)

with isolation of x 1unfolding

(1.14)

– a result compatible with Eq. (1.7), once the two fractions are lumped, a 1,2 a 2,1 b 1canceled out with its negative afterward, and a 1,1finally 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).

If a relationship between two real variables, y and x , is such that y becomes determined whenever x is given, then y is said to be a univariate (real‐valued) function of (real‐variable) x ; this is usually denoted as y ≡ y { x }, where x is termed independent variable and y is termed dependent variable. The same value of y may be obtained for more than one value of x , but no more than one value of y is allowed for each value of x . If more than one independent variable exist, say, x 1, x 2, …, x n, then a multivariate function arises, y ≡ y { x 1, x 2, …, x n, }. The range of values of x for which y is defined constitutes its interval of definition, and a function may be represented either by an (explicit or implicit) analytical expression relating y to x (preferred), or instead by its plot on a plane (useful and comprehensive, except when x grows unbounded) – whereas selected values of said function may, for convenience, be listed in tabular form.