letters
using the row operations indicated, to the equivalent 5 X 6 matrix shown. This indicates that one equation (M - R = 5 - 4 = I), in the coefficients vl, v2, va, vr, v5, v6, is discarded, leaving four linearly independent equations (C - R 1 = 6 -4 1 = 3) each (in this case) involving three coefficients. This example satisfies Roger Crocker's condition n' > n (6 > 5 ) , so that, according to his account, elimination of unknowns should give a linear equation in n' - n 1 or 2 unknowns, which is not so. Counter examples can also be given of Roger Crocker's statements concerning systems for which n' I n. It is perhaps worth remarking that the row operation fnethod just outlined, is entirely equivalent to the linear algebraic matrix method for making chemical equations [(COPLEY, G. N., Chemistry, 41(9), 22 (1968) I.
+
+
Algebra and Chemical Equations
T o the Editor: If by "elimination of the unknowns" Roger Crocker [J. CHEM.EDUC., 45,731 (1968)l means the reduction of a system of n lmear equations in n' unknown stoichiometric coefficients using row operations (Corn, P. M., "Linear Equations," Routledge and Kegan Paul, London, 1958), then it can be shown that some of the statements he makes in his article are not generally true. Employing notation previously adopted [COPLEY, G. N., The School Science Review, 44,744 (1963) 1, if R is the rank of an M X C chemical formulas matrix, then it is possible, using row operations, to convert it into an M X C matrix of the partitioned diagonal form a n h l . . .alRalR+I. . .a10 aslarr.. .amam+1. . .am ....... . ... .. .. . ..... r(1" aRLaR1.
.
. .aRRaRR+l. .aRC
. . ... . . , . , , . . . . . . . . . . . . . .. amam. . . ~ M R ~ M R. .+aLm.
I
operations
CIS0 . . . 0 0 CSS.. . O
CIR+I.
0 0 . . .o
0 . . .o
C,a+,.
. .C,c . .Cse
This form corresponds to M - R of the M equations in C stoichiometric coefficients vlrv27. . .,uc being discarded, leaving R linearly independent equations
It is seen that each of these equations involves not more 1 stoichiometric coefficients. For exthan C - R
+
then, using r to signify row, the initial 5 X 6 formulas matrix is transformed
I 1--
2 0 0 0 6 4 TS-~I 0 1 0 0 3 2 r2-m
Na 0 0 0
2 2 0 0 rs 1 2 0 4 3 2 r,
+ r* + ri - r.12 - r, + n/4
rl/4 -1114
T o the Editor: I n reply to Mr. Copley's letter, the author would like to make the following comments. The author, being a mathematician, was well aware that the statement that a system of n equations in n' unknowns is reducible to a single equation inn' - n 1 unknowns when n' > n has occasional exceptions. However, it is almost always true; and more importanttrue for almost any chemical equation likely to occur in reality. (Mr. Copley's constructed exception is very artificial to say the least.) I n applications, this is what really matters. It should be added that the author's method will work perfectly well even in the rare cases where the above statement does not exactly hold. Thus, Mr. Copley's objection is quite academic. If n' ,< n, however, then a correction must be made. The author does not know whether this was the correction Mr. Copley had in mind since his statement is so vague. The systems of equations here, of course, contain no constant term (and are thus said to be homogeneous). Then, if such a system has one solution in positive integers (which the systems in this application, being physically realizable, always have), it will have an infinity of them. If the unknowns are eliminated, one will find that one is left with an equation of the form ax = by, handled exactly as described at the end of example I1 in the article. Because of the implications of the first and last paragraphs in Mr. Copley's letter, it is only fair to point out that the author's method is actually quite different from Mr. Copley's matrix method in that diophantiue techniques are actually applied, and no use a t all is made of any row operation methods. (In the author's opinion the diophantine method is far easier to use.) The matrix method is not a t all applicable to the second application in the author's paper (The Finding of the Molecular Formula).
+
a ~ + u a ~ + . ~~R~+.I R ~ R + I ~ .R ~+ R+ L .L C
H C
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Volume 46, Number 10, October 1969
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699
