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

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target="_blank" rel="nofollow" href="#ulink_f2b317dd-84b0-5b78-9ae3-f88bdb3537a7">(4.124)

      therefore, if A⁻¹ is described by

(4.125)

then one may insert Eq. (4.125) and Eq. (4.1) with m = n to transform Eq. (4.124) to

(4.126)

– corresponding to AA⁻¹ being equal to I_n . Recalling the algorithm of multiplication of matrices as per Eq. (4.47), one finds that Eq. (4.126) is equivalent to

(4.127)

      hence, a system of n² linear algebraic equations in n² unknowns, i.e. α_1,1, α_1,2, …, α_1,n, α_2,1, α_2,2, …, α_2,n, …, α_n,1, α_n,2, …, α_n,n, arises – the solution of which will be postponed at this point.

If one labels

(4.128)

      then Eq. (4.124) will read

(4.129)

in terms of left‐ and right‐hand sides; this is equivalent to Eq. (4.127), as seen above. Assume now that another matrix C exists, such that

(4.130)

      thus mimicking the intermediate and right‐hand sides of Eq. (4.124); in view of Eq. (4.64), one has it that

      (4.131)

where insertion of Eq. (4.130) unfolds

      (4.132)

      After applying the associative property as conveyed by Eq. (4.57), one gets

      (4.133)

where combination with Eq. (4.129) gives rise to

      (4.134)

      and finally to

      (4.135)

on account of Eqs. (4.61) and (4.128). In other words, the only matrix C that satisfies Eq. (4.130) is indeed A⁻¹, to be calculated via Eq. (4.127), so the second equality in Eq. (4.124) is fully proven once the first equality is true; this result is also consistent with the need that the number of columns of A matches the number of rows of A⁻¹ (thus guaranteeing existence of AA⁻¹) and vice versa (so as to assure existence of A⁻¹ A) – which obviously implies that A and A⁻¹ are square matrices of similar order.

      In view of the definition of inverse, one realizes that

(4.136)

based on Eq. (4.124) after replacing A by A⁻¹ – so ordered subtraction of Eq. (4.136) from Eq. (4.124) gives rise to

(4.137)

      once the left‐ and middle‐hand sides of the former have been previously swapped; postmultiplication of the first equality in Eq. (4.137) by A produces

      (4.138)

where Eqs. (4.70) and (4.76) permit conversion to

(4.139)

The second equality in Eq. (4.124) then supports transformation of Eq. (4.139) to

      (4.140)

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