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Determinantia: proprietates et regvla Crameri

Svmmarivm

Haec lectio ad matrices qvadratas cvivsvis ordinis evm nvmervm extendit qvi, in lectione de systematibvs linearibvs cvrsvs Algebra et Geometria I, decernebat vtrvm systema dvarvm aeqvationvm cvm dvabvs incognitis vnicam solvtionem habeat. Determinans ordinis secvndi interpretatvr vt aream orientatam parallelogrammi qvod colvmnae matricis determinant; determinans ordinis tertii per evolvtionem secvndvm primvm versvm definit et ostendit id cvm regvla Sarri congrvere; et determinans ordinis nn per recvrsionem definit. Per indvctionem in ordinem, ab hac definitione profecta, proprietates determinantis demonstrat (linearitatem in qvoqve versv, permvtationem versvvm, versvs nvllos vel aeqvales, additionem mvltipli alivs versvs, transpositam et matrices triangvlares), easqve ad calcvlvm per redvctionem et ad nexvm inter determinans et rangvm adhibet. Theorema determinantis prodvcti in ordine secvndo demonstrat, cvivs demonstratio generalis in lectione de matrice inversa datvr, et regvlam Crameri in omni ordine; problemata determinantia adhibent ad aream triangvli, ad determinans Vandermondii, ad identitatem Cassinianam nvmerorvm Fibonacci et ad circvmferentiam per tria pvncta transevntem.

Proposita discendi

  1. Determinantia ordinis secvndi et tertii per evolvtionem secvndvm versvm et per regvlam Sarri compvtare, et determinans ordinis secvndi vt aream orientatam interpretari.
  2. Proprietates determinantis per indvctionem in ordinem demonstrare atqve eis vti ad determinantia cvivsvis ordinis compvtanda per redvctionem ad matricem triangvlarem.
  3. Determinans cvm rango et cvm prodvcto matricvm conferre, et eo decernere vtrvm matrix singvlaris sit.
  4. Systemata nn aeqvationvm cvm nn incognitis per regvlam Crameri solvere, et determinantia ad problemata geometriae analyticae et seriei nvmerorvm adhibere.

Determinans ordinis secvndi et area orientata

In lectione cvrsvs I de aeqvationibvs primi gradvs et systematibvs linearibvs nvmervs D=a1b2−a2b1D = a_1 b_2 - a_2 b_1 determinans systematis dvarvm aeqvationvm appellatvs est, et regvla Crameri demonstrata est: si DD nvllvm non est, systema vnam prorsvs solvtionem habet. In lectione de vectoribvs in plano idem nvmervs, ex componentibvs dvorvm vectorvm formatvs, decernebat vtrvm paralleli essent. Iam vero in lectione de matricibvs et operationibvs cvm matricibvs matrices qvadratae introdvctae svnt, et in lectione de methodo Gavssii et Iordani omne systema lineare in forma AX=BAX = B scriptvm est. Qvod si ita est, convenit illvm nvmervm vt fvnctionem matricis coefficientivm considerare, evmqve ad matrices qvadratas cvivsvis ordinis extendere.

Determinans matricis qvadratae AA ordinis secvndi est prodvctvm elementorvm diagonalis principalis demptvm prodvcto elementorvm alterivs diagonalis. Scribitvr det⁡A\det A, avt elementis matricis inter dvas lineolas verticales positis.
A=(abcd),det⁡A=∣abcd∣=ad−bcA = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad \det A = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Determinans systematis dvarvm aeqvationvm est id matricis coefficientivm eivs, qvae in prima colvmna coefficientes ipsivs xx habet et in secvnda eos ipsivs yy: eivs determinans est a1b2−a2b1=Da_1 b_2 - a_2 b_1 = D. Lineolae hic valorem absolvtvm non significant, qvia determinans negativvm esse potest; exempli gratia:

∣3112∣=6−1=5,∣1321∣=1−6=−5\begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} = 6 - 1 = 5, \qquad \begin{vmatrix} 1 & 3 \\ 2 & 1 \end{vmatrix} = 1 - 6 = -5

Dvae colvmnae matricis ordinis secvndi svnt vectores plani, et theorema areae orientatae, infra envntiatvm, determinanti significationem geometricam tribvit qvae obiter criterivm parallelismi lectionis cvrsvs I de vectoribvs explicat.

TheoremaArea orientata

Sint u⃗=(u1,u2)\vec{u} = \left( u_1, u_2 \right) et v⃗=(v1,v2)\vec{v} = \left( v_1, v_2 \right) vectores plani, et sit AA matrix cvivs colvmnae svnt u⃗\vec{u} et v⃗\vec{v}. Area parallelogrammi qvod sagittae ipsorvm u⃗\vec{u} et v⃗\vec{v} ab eadem origine dvctae determinant est valor absolvtvs ipsivs det⁡A\det A. Praeterea, si vectores neqve nvlli neqve paralleli svnt, det⁡A\det A positivvm est cvm conversio minor dimidio circvitv qvae directionem ipsivs u⃗\vec{u} in directionem ipsivs v⃗\vec{v} transfert contra motvm horologii fit, et negativvm cvm secvndvm motvm horologii fit.

det⁡A=∣u1v1u2v2∣=u1v2−u2v1\det A = \begin{vmatrix} u_1 & v_1 \\ u_2 & v_2 \end{vmatrix} = u_1 v_2 - u_2 v_1

Si u⃗\vec{u} nvllvm est, parallelogrammvm in segmentvm contrahitvr et determinans nvllvm est; ponimvs igitvr u⃗\vec{u} nvllvm non esse. Demonstratio vtitvr distantia pvncti a recta, in lectione cvrsvs I de vectoribvs inventa, coordinatis polaribvs lectionis de coordinatis polaribvs et formvla sinvs differentiae lectionis de formvlis additionis et svbtractionis. Figvra eam comitatvr cvm u⃗=(3,1)\vec{u} = \left( 3, 1 \right) et v⃗=(1,2)\vec{v} = \left( 1, 2 \right).

Demonstratio

  1. n⃗=(−u2,u1),n⃗⋅u⃗=−u2u1+u1u2=0,∥n⃗∥=∥u⃗∥\vec{n} = \left( -u_2, u_1 \right), \qquad \vec{n} \cdot \vec{u} = \resaltar{-u_2 u_1 + u_1 u_2} = \resaltar{0}, \qquad \dato{n}{\norma{\vec{n}} = \norma{\vec{u}}}

    Admittitvr, vt in geometria elementari, aream parallelogrammi esse prodvctvm basis et altitvdinis respondentis. Pro basi svmimvs latvs qvod u⃗\vec{u} repraesentat, in recta per originem cvm ea directione transevnte; eivs aeqvatio est −u2x+u1y=0-u_2 x + u_1 y = 0, cvi origo et extremvm ipsivs u⃗\vec{u} satisfacivnt. Per theorema vectoris normalis, vector n⃗\vec{n} ad eam rectam perpendicvlaris est, et eivs norma est norma ipsivs u⃗\vec{u}.

  2. h=∣v⃗⋅n⃗∣∥n⃗∥=∣u1v2−u2v1∣∥n⃗∥h = \frac{\abs{\resaltar{\vec{v} \cdot \vec{n}}}}{\norma{\vec{n}}} = \dato{h}{\frac{\abs{\resaltar{u_1 v_2 - u_2 v_1}}}{\norma{\vec{n}}}}

    Altitvdo est distantia extremi ipsivs v⃗\vec{v} ab ea recta. In formvla distantiae originem vt pvnctvm rectae svmimvs, ita vt vector qvi a recta ad pvnctvm it sit ipse v⃗\vec{v}.

  3. ∥u⃗∥⋅h=∣u1v2−u2v1∣=∣det⁡A∣,∣3⋅2−1⋅1∣=5\resaltar{\norma{\vec{u}}} \cdot h = \resaltar{\abs{u_1 v_2 - u_2 v_1}} = \abs{\det A}, \qquad \abs{3 \cdot 2 - 1 \cdot 1} = 5

    Area est basis in altitvdinem dvcta, et norma ipsivs u⃗\vec{u} cvm norma ipsivs n⃗\vec{n} tollitvr. In figvra basis 10\sqrt{10} metitvr, altitvdo 510\frac{5}{\sqrt{10}}, et area est 55.

  4. det⁡A=(rcos⁡α)(ssin⁡β)−(rsin⁡α)(scos⁡β)=rs(sin⁡βcos⁡α−cos⁡βsin⁡α)=rssin⁡(β−α)\det A = (r \cos \alpha)(s \sin \beta) - (r \sin \alpha)(s \cos \beta) = rs \resaltar{(\sin \beta \cos \alpha - \cos \beta \sin \alpha)} = \dato{s}{rs \resaltar{\sin(\beta - \alpha)}}

    Ad signvm, vectores in coordinatis polaribvs scribimvs: rr et ss svnt eorvm normae, et α\alpha et β\beta angvli qvos cvm semiaxe positivo abscissarvm efficivnt, ita vt u⃗=(rcos⁡α,rsin⁡α)\vec{u} = \left( r \cos \alpha, r \sin \alpha \right) et v⃗=(scos⁡β,ssin⁡β)\vec{v} = \left( s \cos \beta, s \sin \beta \right). Per formvlam sinvs differentiae, determinans est rsrs in sinvm angvli qvi a u⃗\vec{u} ad v⃗\vec{v} it dvctvm.

  5. ∣3112∣=5,∣1321∣=−5\begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} = \resaltar{5}, \qquad \begin{vmatrix} 1 & 3 \\ 2 & 1 \end{vmatrix} = \resaltar{-5}

    Cvm rs>0rs > 0, determinans signvm ipsivs sin⁡(β−α)\sin(\beta - \alpha) habet, qvod positivvm est si β−α\beta - \alpha inter 0∘0^\circ et 180∘180^\circ iacet, mvltiplo ipsivs 360∘360^\circ neglecto, id est si v⃗\vec{v} ab u⃗\vec{u} attingitvr contra motvm horologii minvs dimidio circvitv convertendo, et negativvm in casv contrario. Nvllvm est si et solvm si β−α\beta - \alpha mvltiplvm ipsivs 180∘180^\circ est, id est si vectores paralleli svnt. In figvra, β−α=45∘\beta - \alpha = 45^\circ; colvmnis permvtatis, conversio qvae a v⃗\vec{v} ad u⃗\vec{u} dvcit secvndvm motvm horologii fit, et signvm mvtatvr.

Theorema ostendit ivre determinans ordinis secvndi aream orientatam appellari: eivs valor absolvtvs aream metitvr, et signvm eivs sensvm conversionis qvae a prima colvmna ad secvndam dvcit. Criterivm parallelismi lectionis cvrsvs I de vectoribvs est casvs areae nvllae: dvo vectores paralleli svnt prorsvs cvm parallelogrammvm qvod determinant in rectam complanatvr. Nexvs inter determinans et areas figvrarvm qvarvmlibet, cvm matrix vt transformatio plani agit, in lectione de matricibvs vt transformationibvs plani tractabitvr.