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

The inverse ( n × n ) matrix A −1, of a given ( n × n ) matrix A, satisfies, by definition,

(4.124)

therefore, if A −1is 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 −1being 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 n 2linear algebraic equations in n 2unknowns, 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 Cexists, 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 Cthat 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 Amatches 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 Aand A −1are square matrices of similar order.

In view of the definition of inverse, one realizes that

(4.136)

based on Eq. (4.124)after replacing Aby 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 Aproduces

(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)

after having applied the associative property as per Eq. (4.57)– whereas Eq. (4.61)accounts for simplification to

(4.141)

one thus concludes that

(4.142)

after adding ( A −1) −1to both sides, and recalling Eqs. (4.19)and (4.45). Therefore, composition of the inversion operation with itself cancels it out – in much the same way already found for transposal. A similar reasoning can be developed involving premultiplication of the second equality of Eq. (4.137)by A, viz.

(4.143)

where the distributive property as per Eq. (4.82)and the associative property as per Eq. (4.57), coupled with Eq. (4.67)yield

(4.144)

the definition of inverse labeled as Eq. (4.124)may again be invoked to write

(4.145)