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

or else

(4.146)

in view of Eq. (4.64)– which retrieves Eq. (4.141), and consequently leads also to Eq. (4.142).

On the other hand, one finds that

(4.147)

i.e. the inverse of a product of matrices is given by the product of their inverses, in reverse order; to prove so, one should realize that

(4.148)

can be obtained after postmultiplying ABby B −1 A −1, followed by application of Eq. (4.56)– where both Aand Bare ( n × n ) matrices. In view of Eq. (4.124), one may replace Eq. (4.148)by

(4.149)

where Eqs. (4.57), (4.61), and (4.124)allow further simplification to

(4.150)

one may similarly show that

(4.151)

involving premultiplication of ABby B −1 A −1– again on the basis of the associative property of multiplication of matrices as per Eq. (4.56), which degenerates to

(4.152)

due again to Eq. (4.124). In view of the features of I nas neutral element as conveyed by Eq. (4.64), one may redo Eq. (4.152)to

(4.153)

– again with the aid of Eq. (4.124); the set of Eqs. (4.150)and (4.153)guarantees full validity of Eq. (4.147), in view of the definition of inverse labeled as Eq. (4.124).

The result conveyed by Eq. (4.147)can obviously be extended to any number of factors – by sequentially applying it pairwise, i.e. the inverse of a product of matrices is but the product of their inverses, again in reverse order. When the matrices of interest are identical, this rule leads to

(4.154)

where the right‐hand side may be rewritten as

(4.155)

owing to the definition of power; hence, the power and inverse signs are interchangeable.

In the particular case of matrix Adegenerating to scalar matrix α In , Eq. (4.147)prompts

(4.156)

the inverse of a scalar matrix α I nis given by merely α −1 I n, since I nis the neutral element of multiplication, so Eq. (4.156)is equivalent to

(4.157)

also with the aid of Eq. (4.24)– where Eq. (4.61)supports final transformation to

(4.158)

Equation (4.158)consequently indicates that the inverse of the product of a scalar by a matrix is simply the product of the reciprocal of the said scalar by the inverse of the matrix proper.

One may finally investigate what the combination of the transpose and inverse operators will look like, by first setting the product A T× ( A −1) Tand then realizing that

(4.159)

based on Eq. (4.120); however, Eq. (4.124)has it that

(4.160)

where Eq. (4.108)allows further simplification to

(4.161)

One may similarly write

(4.162)

at the expense again of the rule of transposition of a product of matrices, see Eq. (4.120); the definition of inverse as conveyed by Eq. (4.124)permits simplification of Eq. (4.162)to

(4.163)

whereas Eq. (4.108)may again be invoked to attain

(4.164)

Inspection of Eqs. (4.161)and (4.164)confirms compatibility with the form of Eq. (4.124), so one concludes that

(4.165)

– meaning that the inverse of A Tis merely the transpose of A −1; therefore, the transpose and inverse operators can also be exchanged without affecting the final result.

Although being square is a necessary condition for invertibility of a matrix, it is far from being also a sufficient condition; in fact, the rank of ( n × n ) matrix Amust coincide with its order, so as to guarantee existence of A −1(to be discussed later). Under such conditions, the said square matrix is termed regular – otherwise it is termed singular; as will be seen, the associated determinant is a convenient tool to effect this distinction.