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

(4.177)

Insertion of Eq. (4.177) transforms Eq. (4.168) to

      (4.178)

      where B_1,1 may be factored out as

      (4.179)

      in agreement with Eq. (4.76); isolation of B_1,1 then becomes possible via premultiplication of both sides by ( A_1,1 − A_1,2 A_2,1)⁻¹, i.e.

(4.180)

once again with the aid of Eqs. (4.61) and (4.124). One may likewise combine Eqs. (4.171) and (4.174) to get

      (4.181)

      with the aid of Eq. (4.24), where factoring out of B_2,2 yields

(4.182)

again at the expense of Eq. (4.76), besides Eq. (4.8); after premultiplying both sides by ( A_2,2 − A_2,1 A_1,2)⁻¹, while recalling the definition of inverse as per Eq. (4.124) and the major property of an identity matrix as per Eqs. (4.61) and (4.64), one gets

(4.183)

from Eq. (4.182). In view of Eq. (4.183), one may transform Eq. (4.174) as

      (4.184)

      whereas

(4.185)

results from combination of Eqs. (4.177) and (4.180); therefore, Eqs. (4.180) and (4.183)–(4.185) support reformulation of Eq. (4.87) finally to

      (4.186)

since B = A⁻¹ as per Eqs. (4.124) and (4.166). A similar conclusion on the form of A⁻¹ can, as expected from the double equality in Eq. (4.124), be drawn if AB is replaced by BA in Eq. (4.166).

4.6 Combined Features

      Among the many possible combinations of matrix operations, two of them hold a particular relevance. The first pertains to a symmetric matrix, with regard to pre‐ and postmultiplication by a vector or its transpose – while the other encompasses a related property, which eventually supports definition of a positive semidefinite matrix.

      4.6.1 Symmetric Matrix

      One interesting property of a symmetric (n × n) matrix V, defined as

(4.187)

and consistent with Eq. (4.107), entails the alternate product by vector column a, of length n, viz.

(4.188)

      and vector column b, also of length n, viz.

(4.189)

the product of a^T as per Eq. (4.188) by V as per Eq. (4.187) reads

      (4.190)

      following Eq. (4.47) for the algorithm of multiplication of matrices, coupled with Eq. (4.105) to calculate a^T – whereas a further multiplication of a^T V by b unfolds

(4.191)