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

Equation (4.20)may now be invoked to write

(4.32)

complemented with the distributive property of multiplication of scalars – where application of Eqs. (4.4)and (4.20)leads to

(4.33)

or, once Eqs. (4.3)and (4.9)are taken into account,

(4.34)

Therefore, multiplication of scalar by matrix is distributive, with regard to addition of matrices.

One may conceive a similar property encompassing addition of scalars, i.e.

(4.35)

as per Eq. (4.9)– which becomes

(4.36)

owing to Eq. (4.20), coupled with the distributive property of multiplication of plain scalars. Reverse application of Eq. (4.4)leads to

(4.37)

and an extra utilization of Eq. (4.20)entails

(4.38)

– or else

(4.39)

at the expense of Eq. (4.9). Based on Eq. (4.39), one realizes that multiplication of scalar by matrix is distributive also with regard to addition of scalars.

A final property pertains to product of (scalar) unity by a matrix, according to

(4.40)

based on Eq. (4.2); application of Eq. (4.20)permits, in turn, transformation to

(4.41)

Since 1 is the neutral element of multiplication of scalars, Eq. (4.41)can be redone as

(4.42)

that is equivalent to

(4.43)

in view again of Eq. (4.2)– thus leaving the matrix unchanged, whatever it is; 1 is accordingly confirmed as the neutral element of multiplication of scalar by matrix.

When the scalar at stake is − 1, its product by matrix Atransforms every element thereof to its negative; the result, usually denoted as −A , looks like

(4.44)

as per Eqs. (4.2)and (4.20). Matrix −A is called symmetric of A– because addition of those two matrices produces a nil matrix, i.e.

(4.45)

at the expense of Eqs. (4.2), (4.4), and (4.44).

If ( m × n ) matrix A, or [ a i,j] as per Eq. (4.2), and ( n × p ) matrix B, defined as

(4.46)

are considered – with number of columns of Aequal to number of rows of B, then the said matrices can be multiplied via

(4.47)

the product is an ( m × p ) matrix, with generic element d i,l.

Note that multiplication in reverse order will not be possible unless p = m , due to the matching between number of columns of Band number of rows of Athat would then be required; this example suffices to prove that multiplication of matrices is not commutative. However, even in the case of ( n × n ) matrices Aand Bthat can be multiplied in either order, one gets

(4.48)

en lieu of Eq. (4.47), as well as

(4.49)

with generic element e k,j– written with the aid of the commutative property of multiplication of scalars. The equality of d i,lto e j,kcannot be guaranteed, because the elements chosen for the partial two‐factor products are not the same; for instance, the element positioned in the first row and column of ABlooks like , whereas the corresponding element of BAreads – which is obviously distinct from the former, despite coincidence of (only) the first term.

Consider now three matrices A, B, and C, of the ( m × n ), ( n × p ), and ( p × q ) types, respectively – so product ABis an ( m × p ) rectangular matrix, whereas product of ( m × p ) ABby ( p × q ) Cwill be an ( m × q ) matrix, ABC. Recalling Eqs. (4.2)and (4.46), and complementing them with