determinant 
1572 
determinative 
requires an even or an odd number of transpo- 
sitions to reduce it to the arrangement in the 
square. A determinant is conventionally denoted by 
writing the square block of quantities between two verti- 
cal lines. For example, 
A, B| _ 
a, b\- 
A&y A/5c +0/3C abC + aBc aBy. 
The different products of which a determinant is the sum 
are called its element*. The different quantities which 
are multiplied to form the elements are called the con- 
stituent* of the determinant. The oblique line of places 
from the upper left-hand to the lower right-hand corner 
is called the principal diagonal. The conjugate line of 
places is called the secondary diagonal. The square root 
of the number of constituents is the ordinal number of the 
order or degree of the determinant. Adjugate determi- 
nant one each of whose elements is the cofactor of the 
corresponding term of the determinant to which it is ad- 
jugate. Axlsymmetric determinant. Same as sym- 
metric determinant. See below. Bialar determinant. 
See bialar. Bordered determinant, a determinant 
whose matrix is formed from another by adding new rows 
and columns, especially where a single row and column 
are added, with a zero at their intersection. Centrosym- 
metric determinant, one which is symmetric with re- 
A determinate number of feet. 
Dryden, Essay on Dram. Poesy. 
He talks of power, for example, as if the meaning of the 
word power were as determinate as the meaning of the 
word circle. 
Macaulay, Utilitarian Theory of Government. 
2. Predetermined ; settled ; positive : as, a de- 
terminate rule or order. 
Being delivered by the determinate counsel and fore- 
knowledge of God. Acts 1L 23 - 
3f. Decisive; conclusive. 
I' the progress of this business, 
Ere a determinate resolution, he 
(I mean the bishop) did require a respite. 
Shak., Hen. VIII., H. 4. 
4f. Determined upon; intended. 
My determinate voyage is mere extravagancy. 
SftaK., j.. v.| li. 1. 
5f. Fixed in purpose ; resolute ; determined. 
Like men disused in a long peace ; more determinate to 
do, than skilful how to do. Sir P. Sidney. 
There are some curiosities so bold and determinate as 
to tell the very matter of her prayer. 
Jer. Taylor, Works (ed. 1835), I. 29. 
Determinate Idea, an idea not vague, hut distinguished 
from every other. Determinate individual, in logic, a 
particular individual, designated by name or otherwise, dis- 
roblein. which admits of an infinite number of solutions. 
__ "; ".. With 
in a 
l*AVUAl<u * J **""* .1..-, iem. 111 tIKVIIt,. M11U uL*tyot, n J.I.PI.I. in ......... .....-. - 
a partial determinant, to which it is said to be comple- one s 0]u tj on on i V| or a t i eas t a certain and finite number 
mentary, by having for its constituents all the coustitu- f 80 i ut j ons : | je j ng thus opposed to an indeterminate 
ents of the total determinant which belong to rows and - 
columns from neither of which any constituent of the par- 
tial determinant has teen taken, the sign of the comple- 
mentary determinant being determined by taking its ma- 
trix as it stands in the lower right-hand corner of the ma- 
trix of the total determinant, when the matrix of the par- 
tial determinant has been brought to the upper left-hand 
corner without altering the value of the total determi- 
nant. Composite determinant, a sum of determinants 
whose matrices are obtained by successively omitting all 
the different combinations of n-columns from a rectan- 
gular block of quantities having m-rows and in- and n-col- 
umns. The composite determinant is usually denoted by 
writing its oblong matrix with two vertical lines on each 
side. Compound determinant, a determinant whose 
constituents are themselves determinants. Cubic de- 
terminant, a quantity formed on the analogy of a deter- 
minant proper from a cube of quantities as constituents. 
The principles of religion are ... determinately true 
or false. Tillotlon. 
I have inquired much about Dr. Mead, but can't tell you 
any thing determinately. Walpole., Letters, II. 226. 
We perceive the distance of visible objects more exactly 
and determinately with two eyes than one. 
Reid, Enquiry, vi. 5 22. 
2. Resolutely ; with fixed resolve. 
Determinate^ bent that she would seek all loving means 
to win Zelmane. Sir P. Sidney, Arcadia. 
Before the Reformation, not only were early marriages 
imn;tm propei iium ** I;UWG ^i H u,. l ,im. . v~..ov.v..~...~. determinately discouraged, but the opportunity for them 
Cyclic determinant. Sameascirrafnnt. Determl- did not exist Froude, Sketches, p. 139. 
nant of a linear transformation IT substitution, . , . . t 
the determinant whose constituents are the coefficients determmateness (de-ter ml-nat-nes), n. 1. 
of the equations of transformation regularly arrayed. The state of being determinate, certain, or pre- 
Functional determinant, one in which all the constit- 0jge 
uents in each row are differential coefficients of one quan- 
tity, while all the constituents in each column are differ- 
ential coefficients with respect to one variable. Gauche 
determinant. Same as skew deter minant. See below. 
Minor determinant, or minor of a determinant, a 
On the whole, the variations in the object pursued as 
good . . . have consisted in its acquisition of greater ful- 
ness and dettnninateness. 
T. H. Green, Prolegomena to Ethics, 257. 
determinant whose matrix is formed from the matrix of o ijjjg aua litv of being determined or of per- 
um,r &TS n r!a b S^^^ vow severingVedness of purpose ; determination, 
and one column ; second minor, a minor formed by eras- jjis determinateness and his power seemed to make al- 
ing two rows and two columns, etc. JV-dlmensional lies unnecessary. Jane Austen, Mansfield Park, xiv. 
ue^TSouf to\n h ordinar y a denninant.-0rtho- determination (de-t6r-mi-na'shon), n. [< ME. 
symmetric determinant, one all the constituents of determynation = OF. determinaison, determnot- 
The kynge, by thadvise of his counsell and consent of 
the parties, makethe a fyiiall ende and determynation. 
English Gilds (E. E. T. 8.), p. 305. 
And of the great appearance there was of a speedy de- 
termination of that war. Ludlow, Memoirs, I. 339. 
Zeroaxial determinant. 3 Delimitation; the act of setting bounds to 
nts of the principal diagonal . , , !_!,!_ tho limit* of- mppificallv 
terminant in a narrower sense ol ot determining tne limits mcany, 
determinant, one in which every ^,..,v.v ~. . 
row and j"> column is in every case the negative of the estate. 
one in the j^ row and i* h column, except on the principal 
diagonal. Also called gauche determinant. Skew sym- 
metric determinant, a skew determinant in which all 
the constituents of the principal diagonal vanish. Sym- 
metric determinant, one in which the constituent in 
the i"> row and jth column is in every case equal to that in 
the jth row anil i* column. Zeroaxial determinant, 
one in which the constituent 
are all zeros. [The name dete'mtirtuiti uiuimiiunGi acnac , ~ . - , 
was introduced by Gauss, and was first applied in the assignment to the proper place in a classinca- 
present sense by Cauchy.] tion or series. 
determinantal (de-ter'mi-nan-tal), a. [< de- The pa ,. t jcular determination of the reward or punish- 
terminant + -al.~\ In math., of or pertaining to ment belongeth unto them by whom laws are made, 
determinants. Hooker, Eccles. Polity, i. 10. 
The existence of a notation for the elements of a deter- 3. A determining or deciding, as after consid- 
minantal product and a knowledge of the properties of eration or examination ; specifically, definite or 
the elements facilitate very much the investigation of the authoritative iudicial settlement, as of a con- 
laws of repeated determmanlal multiplication. 
T. Muir, Bipartite Functions, Trans. Royal Soc. of Edin., troversy or suit. 
[XXXII. 478. It may be a question who shal haue the determination 
dptprmiTiatpt ( rip tpr-'mi natl w t K L dp- of such controuersie as may arise whether this or that 
'' v -\ J; T t action or speach be decent or indecent. 
termmatus, pp. of determtnare, limit, fix, deter- Puttenham, Arte of Eng. Poesie, p. 220. 
ShH5 Aeiermin ^ T bring to an end; 4. A decision arrived at or promulgated; an 
authoritative or final ruling; a determinate 
For in every voluntary determination there are certain- 
ly two elements : the consciousness of an energy or effort, 
and a distinct feeling of satisfaction in making the effort. 
Maudsley, Body and Will, p. 87. 
What I affirm is that you have a power of determining 
to act, a power of freely forming the internal act of de- 
termination to do something. 
Mi fart, Xature and Thought, p. 213. 
6. A state of mental decision or resolution with 
regard to some thing; determined purpose; fixed 
intention: as, determination to succeed in an 
enterprise ; his determination was inflexible. 
On the part of the people it [the moral sense] gives rise 
to what we call a jealousy of their liberties a watchful 
determination to resist anything like encroachment upon 
their rights. //. Spencer, Social Statics, p. 266. 
7. The quality of being determined ; fixedness 
of purpose; decision of character; resolute- 
ness : as, a man of determination. 
Violent impulse is not the same as a firm determination. 
J. H. Newman, Parochial Sermons, i. 177. 
8t. In old med., the turning or determining 
point ; the crisis. 
He carefully noted the determination of these maladies. 
Swan, tr. of Sydenham. 
9. Tendency or direction, (a) Of the intellect or 
will toward some object or end by an antecedent mental 
state (idea or motive), determination being in the mental 
what causation is in the physical world. 
Examination is consulting a guide. The determination 
of the will, upon inquiry, is following the direction of that 
guide. Locke, Human Understanding, II. xxi. 50. 
(6) Of the blood : abnormal afflux or flow: as, determina- 
tion of blood to the head. 
10. The solution of a problem, mathematical 
or other ; an ascertainment of any magnitude or 
the value of any quantity; especially, a scien- 
tific evaluation based upon exact physical mea- 
surements: as, a determination of the length 
of the seconds-pendulum. 11. In logic: (a) 
The process of adding characters to a notion, 
and thus rendering it more definite, whether 
this is done by limiting its scope or by an in- 
crease of information. 
This notion, in which ego and non-ego are thought as 
mutually determining, is called by Fichte the category of 
reciprocal determination (Wechselbestimmung). 
Adamson, Fichte, p. 168. 
In the most complete detennination within our reach, 
the conception still does not suffice to enable any one to 
say positively what the perfection of his life would be. 
T. H. Green, Prolegomena to Ethics, 370. 
(ft) The differentiating character itself that is 
added in this process. 
The different determinations of a substance, which are 
nothing but particular modes in which it exists, are called 
accidents. Kant, tr. by Max Miiller. 
12. [ML. determinatioqueEStionte, the answering 
a question, the posting of theses to be defend- 
ed.] In Oxford and other old universities: (a) 
A solemn disputation in which the respondent 
is a bachelor of arts, and which is preparatory 
to graduation as master of arts. (6) A disqui- 
sition or other act substituted in recent times 
for the old disputation. The determinations were 
kept in Lent, and hence often called the Lent determina- 
tions. Originally, in the University of Pads (the model 
of most of the old universities of northern Europe, and 
especially of Oxford and Cambridge), there was but one 
degree, that of master of arts, carrying with it the right to 
lecture regularly in the university. The purpose of the 
determinations was to enable the masters to judge whether 
the candidate was flt to be presented to the chancellor as 
candidate for the mastership; and since there were no 
examinations, there was no other regular means of ascer- 
taining the candidate's fitness. The baccalaureate was 
at first called the determinate, and was originally not a 
degree, nor conferred by the university, but merely a per- 
mission to determine or act as chief respondent in the 
Lent disputations, and was conferred by the "nation." 
In consequence of this inseparable connection between 
the baccalaureate and the determinations, the latter are 
often considered as conditions of the former, although 
they follow in time. 
Hence 13f. A discussion of a question ac- 
cording to the scholastic method, after the 
model of a disputation. 
The sly-slow hours shall not determinate 
The dateless limit of thy dear exile. 
Shak., Rich. II., i. 3. 
opinion or conclusion. 
His [the Mufti's] authoritie is so esteemed that the Em- 
perour will neuer alter a determination made by him. 
Purchas, Pilgrimage, p. 312. 
determinate (de-ter'mi-nat), a. [< ME. deter- 
minat= F. determine = Sp. Pg. determinate ,= j have thu hour lwe[ved g de from our resi( , e|lt 
It. aeternnnato, < L. determiiiatun, pp. : see the with tne determination of the republic on that point. 
verb.] 1. Having defined limits ; fixed; defi- Sterne, Tristram Shandy, iv. 21. 
Another diversity of Method, which is likewise of great 
weight, is the handling of knowledge by ... Questions 
and their Determinations, the latter kind whereof, if it be 
immoderately followed, is as prejudicial to the proceed- 
ing of learning as it is to the proceeding of an army to go 
about to besiege every little fort or hold. 
Bacon, Advancement of Learning, ll. 
He [Wyclif ) broached some singular opinions on several 
abstruse points of metaphysics, which led to determina- 
tions or treatises being published agalint him. 
Encyc. Brit., VIII. 411. 
= Syn. 3. Conclusion, settlement, termination. 7. Reso- 
lution, etc. (sec deciuivn). firmness. 
