By Silvestre François Lacroix

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11) where Ers is the elementary matrix of ord~r (m X n) (the order of X) defined in section 1. 2. r =E~. 13) = [xii] is of order (m X n) A ::: [aii] is or order (l X m) and B = (h i/ ] is of order (n X q) Y = (Yij] is of ordeF (I X q) . A and B are assumed independent of X. Our aim is to find the rule for obtaining the derivatives ay and where X rs Is a typical elellll:nt of X and YI/Is a typicul element of Y. 13) as a function of the elements of X. We can achieve this objective in a number of different ways.

6 PROBLEM 4 We consider a problem similar to the previous one but in a different context. An important concept in Control Theory is the transition matrix. 13) Sec. J If A is a constant matrix, it is easily shown lhal *. = B'(p. . 11) ~ = GifJ . if;(t, t) = *

14) to obtain an expression for aYjox,s. 19) where E/j is an elementary matrix of order (l X q). 15) to write Ut aJ,bsl a2,bsl al,bSl a2,bs2 al,bsq a2,bsq am,bsl am,bSl amrbsq .. ax,s :;: ai" al, [bsl bs2 ... ' :::: ae,e~b 5 find derivative ay ox rs , y="=" ax'b order of a, b is thallhc product on right hand side defined. solution l3y method used above to obtain il ilxr$ (axb), we a - (ax'b)="=" ae~sb axrs '. before continuing with further examples need rule for determining matrices. < p>

### Elements of Algebra by Silvestre François Lacroix

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