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

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(4.156)

the inverse of a scalar matrix α I_n is given by merely α⁻¹ I_n, since I_n is 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)^T and 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^T is 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 A must 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.

      4.5.2 Block Matrix

Oftentimes, matrix A to be inverted appears as a block matrix, in agreement with Eq. (4.86) – so the question is to find its inverse, say, B, in a form consistent with Eq. (4.87); A_1,1 and B_1,1 will hereafter denote regular (m × m) matrices, A_1,2 and B_1,2 denote (m × p) matrices, A_2,1 and B_2,1 denote (p × m) matrices, and A_2,2 and B_2,2 denote regular (p × p) matrices. Under these conditions, Eq. (4.124) may be reformulated to

(4.166)

      so Eq. (4.88) may be retrieved to allow transformation of Eq. (4.166) to

      (4.167)

this is equivalent to writing

(4.168)

(4.169)

(4.170)

      and

(4.171)

owing to the required equality of matrices in the two sides. Equation (4.169) may be rewritten as

      (4.172)

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