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

(4.7)

from Eq. (4.6), and finally

(4.8)

as per Eqs. (4.2)and (4.3)– thus confirming the initial statement.

If a third matrix Cis defined as

(4.9)

then one can write

(4.10)

together with Eqs. (4.2)and (4.3); based on Eq. (4.4), one has that

(4.11)

and a further utilization of Eq. (4.4)leads to

(4.12)

– along with the associative property borne by addition of scalars. One may repeat the above reasoning by first associating Aand B, viz.

(4.13)

at the expense of Eqs. (4.2), (4.3), and (4.9), with Eq. (4.4)allowing transformation to

(4.14)

supplementary use of Eq. (4.4)unfolds

(4.15)

with the aid of the associative property of addition of scalars, while elimination of the right‐hand side between Eqs. (4.12)and (4.15)gives rise to

(4.16)

– meaning that addition of matrices is associative.

For every ( m × n ) matrix A, there is a null matrix 0 m×nsuch that

(4.17)

in agreement with Eq. (4.2), where Eq. (4.4)prompts transformation to

(4.18)

in view of 0 being the neutral element for addition of scalars; Eq. (4.18)finally gives rise to

(4.19)

again at the expense of Eq. (4.2). Therefore, 0 m×nplays the role of neutral element with regard to addition of matrices, i.e. it leaves the other ( m × n ) matrix (to which it is added) unchanged.

Given a generic scalar, say, α , another operation can be defined encompassing matrix Bof any type, viz.

(4.20)

with the aid of Eq. (4.3); this is termed multiplication of scalar by matrix. In view of Eqs. (4.3)and (4.20), one has that

(4.21)

which may be rewritten as

(4.22)

due to the commutativity of addition of scalars – or, upon use of Eq. (4.20)backward,

(4.23)

Eq. (4.3)may again be recalled to write

(4.24)

known as commutative property of multiplication of scalar by matrix – even though the scalar is normally placed up front relative to the matrix, for a matter of convention.

If a second scalar is invoked, say, β , then Eq. (4.9)supports

(4.25)

where application of Eq. (4.20)unfolds

(4.26)

a second application of Eq. (4.20)yields

(4.27)

together with the associative property of multiplication of scalars. Final backward application of Eq. (4.20)gives rise to

(4.28)

or, equivalently,

(4.29)

due to Eq. (4.9); one usually refers to Eq. (4.29)as associative property.

If addition of matrices and multiplication of scalar by matrix are considered simultaneously, then one gets

(4.30)

as per Eqs. (4.3)and (4.9), with Eq. (4.4)supporting transformation to

(4.31)