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

(4.50)

one gets

(4.51)

– where application of the algorithm conveyed by Eq. (4.47)leads to

(4.52)

A second application of the said algorithm to Eq. (4.52)gives rise to

(4.53)

which may be algebraically rearranged as

(4.54)

as per the distributive property of multiplication of plain scalars – where exchange of summations is possible, as no constraint is imposed upon their limits (i.e. n is independent of p ) besides commutativity of addition of scalars; further manipulation yields

(4.55)

at the expense of the algorithm conveyed by Eq. (4.47)applied twice reversewise – thus prompting the conclusion

(4.56)

based on Eqs. (4.2), (4.46), and (4.50). Therefore, multiplication of matrices is associative, provided that the relative order of multiplication of the original factors is kept; Eq. (4.56)is often coined as

(4.57)

in view of the common (intermediate) form in Eq. (4.54).

If ( m × n ) matrix Ais multiplied by the ( n × n ) identity matrix, I n, then Eq. (4.47)can be revisited as

(4.58)

where the identity matrix is defined as

(4.59)

– with a main diagonal of 1's, and 0's elsewhere; since the summations are both nil for carrying a nil factor, Eq. (4.58)breaks down to

(4.60)

– so Eq. (4.2)will finally support

(4.61)

since 1 ≤ r ≤ n . In other words, multiplication of a matrix by the (compatible) identity matrix leaves the former unchanged – so I nplays the role of neutral element for the multiplication of matrices. This very same conclusion can be attained if the order of multiplication is reversed, i.e.

(4.62)

as per Eq. (4.47), provided that the matrices are still compatible with regard to multiplication – so I nhas been swapped with I m; since the right‐hand side reduces to its middle term, Eq. (4.62)simplifies again to

(4.63)

or else

(4.64)

in view of Eq. (4.2)– i.e. the order of multiplication of the identity matrix by another matrix (when feasible) does not affect the final result.

When an ( m × n ) matrix Ais postmultiplied by a (compatible) null ( n × p ) matrix, 0 n×p, one gets

(4.65)

as per Eq. (4.2)– where Eq. (4.47)can be employed to get

(4.66)

together with the trivial rules of multiplication of a plain scalar by zero and summation of any number of resulting zeros; Eq. (4.66)is thus equivalent to

(4.67)

meaning that postmultiplication by the null matrix always degenerates to a null matrix (with the same number of columns). By the same token, premultiplication of Aby the (compatible) null ( p × m ) matrix 0 p×m, i.e.

(4.68)

on the basis of Eq. (4.2), gives rise to

(4.69)

by virtue of Eq. (4.47)– complemented by the nil product of zero by any scalar and the nil sum of resulting zeros; Eq. (4.69)is an alias of

(4.70)

meaning that premultiplication by a null matrix leads necessarily to the corresponding null matrix as product – where the latter has number of columns not necessarily coincident with that of the factor null matrix.

A final property of interest pertains to simultaneous performance of addition and multiplication of matrices, viz.