Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata
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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,
One of the most powerful applications of matrices is in solving sets of linear algebraic equations, say,
and
in its simplest version – where a1,1, a1,2, a2,1, a2,2, b1, and b2 denote real numbers, and x1 and x2 denote variables; if a1,1 ≠ 0 and a1,1 a2,2 − a1,2 a2,1 ≠ 0, then one may start by isolating x1 in Eq. (1.1) as
and then replace it in Eq. (1.2) to obtain
After factoring x2 out, Eq. (1.4) becomes
(1.5)
so isolation of x2 eventually gives
– which yields a solution only when a1,1 a2,2 − a1,2 a2,1 ≠ 0; insertion of Eq. (1.6) back in Eq. (1.3) yields
thus justifying why a solution for x1 requires a1,1 ≠ 0, besides a1,1 a2,2 − a1,2 a2,1 ≠ 0 (as enforced from the very beginning). Equation (1.6) may be rewritten as
(1.8)
– provided that one defines
complemented with
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 −a1,2/a2,2, one would get
– so ordered addition of Eqs. (1.1) and (1.11) produces simply
after having x1 factored out; upon multiplication of both sides by a2,2, Eq.,(1.12) becomes
(1.13)
with isolation of x1 unfolding
– a result compatible with Eq. (1.7), once the two fractions are lumped, a1,2 a2,1 b1 canceled out with its negative afterward, and a1,1 finally dropped from numerator and denominator. Recalling Eq.