Skip to main content Contents Prev Up Next \(\def\vs{{\it vs. }}
\def\cf{{\it cf. }}
\def\viz{{\it viz. }}
\def\ie{{\it i.e. }}
\def\etc{{\it etc. }}
\def\eg{{\it e.g. }}
\def\etal{{\it et al .}}
\def\via{{\it via }}
\def\adhoc{{\it ad hoc }}
\def\apriori{{\it apriori }}
\def\Afrak{\mathfrak{A}}
\def\Bfrak{\mathfrak{B}}
\def\Cfrak{\mathfrak{C}}
\def\Dfrak{\mathfrak{D}}
\def\Efrak{\mathfrak{E}}
\def\Ffrak{\mathfrak{F}}
\def\Gfrak{\mathfrak{G}}
\def\Hfrak{\mathfrak{H}}
\def\Ifrak{\mathfrak{I}}
\def\Jfrak{\mathfrak{J}}
\def\Kfrak{\mathfrak{K}}
\def\Lfrak{\mathfrak{L}}
\def\Mfrak{\mathfrak{M}}
\def\Nfrak{\mathfrak{N}}
\def\Ofrak{\mathfrak{O}}
\def\Pfrak{\mathfrak{P}}
\def\Qfrak{\mathfrak{Q}}
\def\Rfrak{\mathfrak{R}}
\def\Sfrak{\mathfrak{S}}
\def\Tfrak{\mathfrak{T}}
\def\Ufrak{\mathfrak{U}}
\def\Vfrak{\mathfrak{V}}
\def\Wfrak{\mathfrak{W}}
\def\Xfrak{\mathfrak{X}}
\def\Yfrak{\mathfrak{Y}}
\def\Zfrak{\mathfrak{Z}}
\def\afrak{\mathfrak{a}}
\def\bfrak{\mathfrak{b}}
\def\cfrak{\mathfrak{c}}
\def\dfrak{\mathfrak{d}}
\def\efrak{\mathfrak{e}}
\def\ffrak{\mathfrak{f}}
\def\gfrak{\mathfrak{g}}
\def\hfrak{\mathfrak{h}}
\def\ifrak{\mathfrak{i}}
\def\jfrak{\mathfrak{j}}
\def\kfrak{\mathfrak{k}}
\def\lfrak{\mathfrak{l}}
\def\mfrak{\mathfrak{m}}
\def\nfrak{\mathfrak{n}}
\def\ofrak{\mathfrak{o}}
\def\pfrak{\mathfrak{p}}
\def\qfrak{\mathfrak{q}}
\def\rfrak{\mathfrak{r}}
\def\sfrak{\mathfrak{s}}
\def\tfrak{\mathfrak{t}}
\def\ufrak{\mathfrak{u}}
\def\vfrak{\mathfrak{v}}
\def\wfrak{\mathfrak{w}}
\def\xfrak{\mathfrak{x}}
\def\yfrak{\mathfrak{y}}
\def\zfrak{\mathfrak{z}}
\def\AA{\mathbb{A}}
\def\BB{\mathbb{B}}
\def\CC{\mathbb{C}}
\def\DD{\mathbb{D}}
\def\EE{\mathbb{E}}
\def\FF{\mathbb{F}}
\def\GG{\mathbb{G}}
\def\HH{\mathbb{H}}
\def\II{\mathbb{I}}
\def\JJ{\mathbb{J}}
\def\KK{\mathbb{K}}
\def\LL{\mathbb{L}}
\def\MM{\mathbb{M}}
\def\NN{\mathbb{N}}
\def\OO{\mathbb{O}}
\def\PP{\mathbb{P}}
\def\QQ{\mathbb{Q}}
\def\RR{\mathbb{R}}
\def\SS{\mathbb{S}}
\def\TT{\mathbb{T}}
\def\UU{\mathbb{U}}
\def\VV{\mathbb{V}}
\def\WW{\mathbb{W}}
\def\XX{\mathbb{X}}
\def\YY{\mathbb{Y}}
\def\ZZ{\mathbb{Z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\Ap{A^\prime}
\def\Bp{B^\prime}
\def\Cp{C^\prime}
\def\Dp{D^\prime}
\def\Ep{E^\prime}
\def\Fp{F^\prime}
\def\Gp{G^\prime}
\def\Hp{H^\prime}
\def\Ip{I^\prime}
\def\Jp{J^\prime}
\def\Kp{K^\prime}
\def\Lp{L^\prime}
\def\Mp{M^\prime}
\def\Mp{N^\prime}
\def\Op{O^\prime}
\def\Pp{P^\prime}
\def\Qp{Q^\prime}
\def\Rp{R^\prime}
\def\Sp{S^\prime}
\def\Tp{T^\prime}
\def\Up{U^\prime}
\def\Vp{V^\prime}
\def\Wp{W^\prime}
\def\Xp{X^\prime}
\def\Yp{Y^\prime}
\def\Zp{Z^\prime}
\def\ap{a^\prime}
\def\bp{b^\prime}
\def\cp{c^\prime}
\def\dprime{d^\prime}
\def\ep{e^\prime}
\def\fp{f^\prime}
\def\gp{g^\prime}
\def\hp{h^\prime}
\def\ip{i^\prime}
\def\jp{j^\prime}
\def\kp{k^\prime}
\def\lp{l^\prime}
\def\mp{m^\prime}
\def\np{n^\prime}
\def\op{o^\prime}
\def\pp{p^\prime}
\def\qp{q^\prime}
\def\rp{r^\prime}
\def\sp{s^\prime}
\def\tp{t^\prime}
\def\up{u^\prime}
\def\vp{v^\prime}
\def\wp{w^\prime}
\def\xp{x^\prime}
\def\yp{y^\prime}
\def\zp{z^\prime}
\def\App{A^{\prime\prime}}
\def\Bpp{B^{\prime\prime}}
\def\Cpp{C^{\prime\prime}}
\def\Dpp{D^{\prime\prime}}
\def\Epp{E^{\prime\prime}}
\def\Fpp{F^{\prime\prime}}
\def\Gpp{G^{\prime\prime}}
\def\Hpp{H^{\prime\prime}}
\def\Ipp{I^{\prime\prime}}
\def\Jpp{J^{\prime\prime}}
\def\Kpp{K^{\prime\prime}}
\def\Lpp{L^{\prime\prime}}
\def\Mpp{M^{\prime\prime}}
\def\Mpp{N^{\prime\prime}}
\def\Opp{O^{\prime\prime}}
\def\Ppp{P^{\prime\prime}}
\def\Qpp{Q^{\prime\prime}}
\def\Rpp{R^{\prime\prime}}
\def\Spp{S^{\prime\prime}}
\def\Tpp{T^{\prime\prime}}
\def\Upp{U^{\prime\prime}}
\def\Vpp{V^{\prime\prime}}
\def\Wpp{W^{\prime\prime}}
\def\Xpp{X^{\prime\prime}}
\def\Ypp{Y^{\prime\prime}}
\def\Zpp{Z^{\prime\prime}}
\def\app{a^{\prime\prime}}
\def\bpp{b^{\prime\prime}}
\def\cpp{c^{\prime\prime}}
\def\dpp{d^{\prime\prime}}
\def\epp{e^{\prime\prime}}
\def\fpp{f^{\prime\prime}}
\def\gpp{g^{\prime\prime}}
\def\hpp{h^{\prime\prime}}
\def\ipp{i^{\prime\prime}}
\def\jpp{j^{\prime\prime}}
\def\kpp{k^{\prime\prime}}
\def\lpp{l^{\prime\prime}}
\def\mpp{m^{\prime\prime}}
\def\npp{n^{\prime\prime}}
\def\opp{o^{\prime\prime}}
\def\ppp{p^{\prime\prime}}
\def\qpp{q^{\prime\prime}}
\def\rpp{r^{\prime\prime}}
\def\spp{s^{\prime\prime}}
\def\tpp{t^{\prime\prime}}
\def\upp{u^{\prime\prime}}
\def\vpp{v^{\prime\prime}}
\def\wpp{w^{\prime\prime}}
\def\xpp{x^{\prime\prime}}
\def\ypp{y^{\prime\prime}}
\def\zpp{z^{\prime\prime}}
\def\abar{\overline{a}}
\def\bbar{\overline{b}}
\def\cbar{\overline{c}}
\def\dbar{\overline{d}}
\def\ebar{\overline{e}}
\def\fbar{\overline{f}}
\def\gbar{\overline{g}}
\def\ibar{\overline{i}}
\def\jbar{\overline{j}}
\def\kbar{\overline{k}}
\def\lbar{\overline{l}}
\def\mbar{\overline{m}}
\def\nbar{\overline{n}}
\def\obar{\overline{o}}
\def\pbar{\overline{p}}
\def\qbar{\overline{q}}
\def\rbar{\overline{r}}
\def\sbar{\overline{s}}
\def\tbar{\overline{t}}
\def\ubar{\overline{u}}
\def\vbar{\overline{v}}
\def\wbar{\overline{w}}
\def\xbar{\overline{x}}
\def\ybar{\overline{y}}
\def\zbar{\overline{z}}
\def\Abar{\overline{A}}
\def\Bbar{\overline{B}}
\def\Cbar{\overline{C}}
\def\Dbar{\overline{D}}
\def\Ebar{\overline{E}}
\def\Fbar{\overline{F}}
\def\Gbar{\overline{G}}
\def\Hbar{\overline{H}}
\def\Ibar{\overline{I}}
\def\Jbar{\overline{J}}
\def\Kbar{\overline{K}}
\def\Lbar{\overline{L}}
\def\Mbar{\overline{M}}
\def\Nbar{\overline{N}}
\def\Obar{\overline{O}}
\def\Pbar{\overline{P}}
\def\Qbar{\overline{Q}}
\def\Rbar{\overline{R}}
\def\Sbar{\overline{S}}
\def\Tbar{\overline{T}}
\def\Ubar{\overline{U}}
\def\Vbar{\overline{V}}
\def\Wbar{\overline{W}}
\def\Xbar{\overline{X}}
\def\Ybar{\overline{Y}}
\def\Zbar{\overline{Z}}
\def\aunder{\underline{a}}
\def\bunder{\underline{b}}
\def\cunder{\underline{c}}
\def\dunder{\underline{d}}
\def\eunder{\underline{e}}
\def\funder{\underline{f}}
\def\gunder{\underline{g}}
\def\hunder{\underline{h}}
\def\iunder{\underline{i}}
\def\junder{\underline{j}}
\def\kunder{\underline{k}}
\def\lunder{\underline{l}}
\def\munder{\underline{m}}
\def\nunder{\underline{n}}
\def\ounder{\underline{o}}
\def\punder{\underline{p}}
\def\qunder{\underline{q}}
\def\runder{\underline{r}}
\def\sunder{\underline{s}}
\def\tunder{\underline{t}}
\def\uunder{\underline{u}}
\def\vunder{\underline{v}}
\def\wunder{\underline{w}}
\def\xunder{\underline{x}}
\def\yunder{\underline{y}}
\def\zunder{\underline{z}}
\def\Aunder{\underline{A}}
\def\atilde{\widetilde{a}}
\def\btilde{\widetilde{b}}
\def\ctilde{\widetilde{c}}
\def\dtilde{\widetilde{d}}
\def\etilde{\widetilde{e}}
\def\ftilde{\widetilde{f}}
\def\gtilde{\widetilde{g}}
\def\htilde{\widetilde{h}}
\def\itilde{\widetilde{i}}
\def\jtilde{\widetilde{j}}
\def\ktilde{\widetilde{k}}
\def\ltilde{\widetilde{l}}
\def\mtilde{\widetilde{m}}
\def\ntilde{\widetilde{n}}
\def\otilde{\widetilde{o}}
\def\ptilde{\widetilde{p}}
\def\qtilde{\widetilde{q}}
\def\rtilde{\widetilde{r}}
\def\stilde{\widetilde{s}}
\def\ttilde{\widetilde{t}}
\def\utilde{\widetilde{u}}
\def\vtilde{\widetilde{v}}
\def\wtilde{\widetilde{w}}
\def\xtilde{\widetilde{x}}
\def\ytilde{\widetilde{y}}
\def\ztilde{\widetilde{z}}
\def\Atilde{\widetilde{A}}
\def\Btilde{\widetilde{B}}
\def\Ctilde{\widetilde{C}}
\def\Dtilde{\widetilde{D}}
\def\Etilde{\widetilde{E}}
\def\Ftilde{\widetilde{F}}
\def\Gtilde{\widetilde{G}}
\def\Htilde{\widetilde{H}}
\def\Itilde{\widetilde{I}}
\def\Jtilde{\widetilde{J}}
\def\Ktilde{\widetilde{K}}
\def\Ltilde{\widetilde{L}}
\def\Mtilde{\widetilde{M}}
\def\Ntilde{\widetilde{N}}
\def\Otilde{\widetilde{O}}
\def\Ptilde{\widetilde{P}}
\def\Qtilde{\widetilde{Q}}
\def\Rtilde{\widetilde{R}}
\def\Stilde{\widetilde{S}}
\def\Ttilde{\widetilde{T}}
\def\Utilde{\widetilde{U}}
\def\Vtilde{\widetilde{V}}
\def\Wtilde{\widetilde{W}}
\def\Xtilde{\widetilde{X}}
\def\Ytilde{\widetilde{Y}}
\def\Ztilde{\widetilde{Z}}
\def\Alphatilde{\widetilde{\Alpha}}
\def\Betatilde{\widetilde{\Beta}}
\def\Gammatilde{\widetilde{\Gamma}}
\def\Deltatilde{\widetilde{\Delta}}
\def\Epsilontilde{\widetilde{\Epsilon}}
\def\Zetatilde{\widetilde{\Zeta}}
\def\Etatilde{\widetilde{\Eta}}
\def\Thetatilde{\widetilde{\Theta}}
\def\Iotatilde{\widetilde{\Iota}}
\def\Kappatilde{\widetilde{\Kappa}}
\def\Lambdatilde{\widetilde{\Lamdba}}
\def\Mutilde{\widetilde{\Mu}}
\def\Nutilde{\widetilde{\Nu}}
\def\Xitilde{\widetilde{\Xi}}
\def\Omicrontilde{\widetilde{\Omicron}}
\def\Pitilde{\widetilde{\Pi}}
\def\Rhotilde{\widetilde{\Rho}}
\def\Sigmatilde{\widetilde{\Sigma}}
\def\Tautilde{\widetilde{\Tau}}
\def\Upsilontilde{\widetilde{\Upsilon}}
\def\Phitilde{\widetilde{\Phi}}
\def\Chitilde{\widetilde{\Chi}}
\def\Psitilde{\widetilde{\Psi}}
\def\Omegatilde{\widetilde{\Omega}}
\def\alphatilde{\widetilde{\alpha}}
\def\betatilde{\widetilde{\beta}}
\def\gammatilde{\widetilde{\gamma}}
\def\deltatilde{\widetilde{\delta}}
\def\epsilontilde{\widetilde{\epsilon}}
\def\zetatilde{\widetilde{\zeta}}
\def\etatilde{\widetilde{\eta}}
\def\thetatilde{\widetilde{\theta}}
\def\iotatilde{\widetilde{\iota}}
\def\kappatilde{\widetilde{\kappa}}
\def\lambdatilde{\widetilde{\lamdba}}
\def\mutilde{\widetilde{\mu}}
\def\nutilde{\widetilde{\nu}}
\def\xitilde{\widetilde{\xi}}
\def\omicrontilde{\widetilde{\omicron}}
\def\pitilde{\widetilde{\pi}}
\def\rhotilde{\widetilde{\rho}}
\def\sigmatilde{\widetilde{\sigma}}
\def\tautilde{\widetilde{\tau}}
\def\upsilontilde{\widetilde{\upsilon}}
\def\phitilde{\widetilde{\phi}}
\def\chitilde{\widetilde{\chi}}
\def\psitilde{\widetilde{\psi}}
\def\omegatilde{\widetilde{\omega}}
\def\Alphabar{\bar{\Alpha}}
\def\Betabar{\bar{\Beta}}
\def\Gammabar{\bar{\Gamma}}
\def\Deltabar{\bar{\Delta}}
\def\Epsilonbar{\bar{\Epsilon}}
\def\Zetabar{\bar{\Zeta}}
\def\Etabar{\bar{\Eta}}
\def\Thetabar{\bar{\Theta}}
\def\Iotabar{\bar{\Iota}}
\def\Kappabar{\bar{\Kappa}}
\def\Lambdabar{\bar{\Lamdba}}
\def\Mubar{\bar{\Mu}}
\def\Nubar{\bar{\Nu}}
\def\Xibar{\bar{\Xi}}
\def\Omicronbar{\bar{\Omicron}}
\def\Pibar{\bar{\Pi}}
\def\Rhobar{\bar{\Rho}}
\def\Sigmabar{\bar{\Sigma}}
\def\Taubar{\bar{\Tau}}
\def\Upsilonbar{\bar{\Upsilon}}
\def\Phibar{\bar{\Phi}}
\def\Chibar{\bar{\Chi}}
\def\Psibar{\bar{\Psi}}
\def\Omegabar{\bar{\Omega}}
\def\alphabar{\bar{\alpha}}
\def\betabar{\bar{\beta}}
\def\gammabar{\bar{\gamma}}
\def\deltabar{\bar{\delta}}
\def\epsilonbar{\bar{\epsilon}}
\def\zetabar{\bar{\zeta}}
\def\etabar{\bar{\eta}}
\def\thetabar{\bar{\theta}}
\def\iotabar{\bar{\iota}}
\def\kappabar{\bar{\kappa}}
\def\lambdabar{\bar{\lamdba}}
\def\mubar{\bar{\mu}}
\def\nubar{\bar{\nu}}
\def\xibar{\bar{\xi}}
\def\omicronbar{\bar{\omicron}}
\def\pibar{\bar{\pi}}
\def\rhobar{\bar{\rho}}
\def\sigmabar{\bar{\sigma}}
\def\taubar{\bar{\tau}}
\def\upsilonbar{\bar{\upsilon}}
\def\phibar{\bar{\phi}}
\def\chibar{\bar{\chi}}
\def\psibar{\bar{\psi}}
\def\omegabar{\bar{\omega}}
\def\del{\partial}
\def\delbar{\overline{\partial}}
\def\Cech{\check{C}}
\def\half{\frac{1}{2}}
\def\defeq{\mathrel{\mathop:}=}
\def\alg{\mathrm{alg}}
\def\Alt{\mathrm{Alt}}
\def\Amp{\mathrm{Amp}}
\def\Arg{\mathrm{Arg}}
\def\an{\mathrm{an}}
\def\anti{\mathrm{anti}}
\def\Ap{\mathrm{Ap}}
\def\arcsinh{\mathrm{arcsinh\hspace{0.07cm}}}
\def\arccosh{\mathrm{arccosh\hspace{0.07cm}}}
\def\arctanh{\mathrm{arctanh\hspace{0.07cm}}}
\def\arccsch{\mathrm{arccsch\hspace{0.07cm}}}
\def\arcsech{\mathrm{arcsech\hspace{0.07cm}}}
\def\arccoth{\mathrm{arccoth\hspace{0.07cm}}}
\def\arccsc{\mathrm{arccsc\hspace{0.07cm}}}
\def\arcsec{\mathrm{arcsec\hspace{0.07cm}}}
\def\arccot{\mathrm{arccot\hspace{0.07cm}}}
\def\arg{\mathrm{arg}}
\def\BC{\mathrm{BC}}
\def\Bel{\mathrm{Bel}}
\def\calCH{\mathcal{CH}}
\def\csch{\mathrm{csch}\hspace{0.07cm}}
\def\CH{\mathrm{CH}}
\def\ch{\mathrm{ch}}
\def\closed{\mathrm{closed}}
\def\codim{\mathrm{codim}}
\def\coth{\mathrm{coth}\hspace{0.07cm}}
\def\Coh{\mathfrak{Coh}}
\def\Coker{\mathrm{Coker}}
\def\Cone{\mathrm{Cone}}
\def\darg{d\mathrm{arg}}
\def\Db{\mathrm{Db}}
\def\dclosed{\mathrm{d-closed}}
\def\deg{\mathrm{deg}}
\def\dim{\mathrm{dim}}
\def\divisor{\mathrm{div}}
\def\dlog{d\mathrm{log}}
\def\DNE{\mathrm{DNE}}
\def\DR{\mathrm{DR}}
\def\DST{\mathrm{DST}}
\def\exp{\mathrm{exp}}
\def\FLB{\mathrm{FLB}}
\def\FLS{\mathrm{FLS}}
\def\Gr{\mathrm{Gr}}
\def\Hzar{H_{\mathrm{Zar}}}
\def\Hol{\mathrm{Hol}}
\def\Id{\mathrm{Id}}
\def\Image{\mathrm{Im}}
\def\Ka{\mathcal{K}_A}
\def\Ker{\mathrm{Ker}}
\def\kod{\mathrm{kod}}
\def\Kx{\mathcal{K}_X}
\def\Kz{\mathcal{K}_Z}
\def\log{\mathrm{log}}
\def\Log{\mathrm{Log}}
\def\Li{\mathrm{Li}}
\def\min{\mathrm{min}}
\def\Mon{\mathrm{Mon}}
\def\Nef{\mathrm{Nef}}
\def\NS{\mathrm{NS}}
\def\Oa{\mathcal{O}_A}
\def\Ox{\mathcal{O}_X}
\def\Oz{\mathcal{O}_Z}
\def\Perp{\mathrm{Perp}}
\def\Pic{\mathrm{Pic}}
\def\Proj{\mathrm{Proj}}
\def\rank{\mathrm{rank}}
\def\Rat{\mathrm{Rat}}
\def\Real{\mathrm{Re}}
\def\reg{\mathrm{reg}}
\def\Res{\mathrm{Res}}
\def\res{\mathrm{res}}
\def\Ric{\mathrm{Ric}}
\def\sech{\mathrm{sech}\hspace{0.07cm}}
\def\Span{\mathrm{Span}}
\def\Spec{\mathrm{Spec}}
\def\sing{\mathrm{sing}}
\def\Singx{\mathrm{Sing}(X)}
\def\sheafKer{\mathcal{\Ker}}
\def\sheafIm{\mathcal{\Im}}
\def\Span{\mathrm{Span}}
\def\Spin{\mathrm{Spin}}
\def\Str{\mathrm{Str}}
\def\td{\mathrm{td}}
\def\tr{\mathrm{tr}}
\def\Todd{\mathrm{Todd}}
\def\tor{\mathrm{tor}}
\def\trdeg{\mathrm{trdeg}}
\def\Zar{\mathrm{Zar}}
\def\ZFLS{\mathrm{ZFLS}}
\usepackage{tikz}
\usepackage{tkz-graph}
\usepackage{tkz-euclide}
\usetikzlibrary{patterns}
\usetikzlibrary{positioning}
\usetikzlibrary{matrix,arrows}
\usetikzlibrary{calc}
\usetikzlibrary{shapes}
\usetikzlibrary{through,intersections,decorations,shadows,fadings}
\usepackage{pgfplots}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 8.2 Cofactor Expansion
Subsection 8.2.1 \(2 \times 2\) determinants
The determinant of a (square) matrix is constructed recursively, starting with \(2 \times 2\) matrices. Therefore, I need to first show how to calculate the determinant of a \(2 \times
2\) matrix.
\begin{equation*}
\det \left( \begin{matrix} a \amp b \\ c \amp d \end{matrix}
\right) = \left| \begin{matrix} a \amp b \\ c \amp d
\end{matrix} \right| = ad-bc
\end{equation*}
Example 8.2.1 .
For the matrix \(\left( \begin{matrix}1 \amp -3 \\ 2 \amp -1
\end{matrix} \right)\text{,}\) the determinant is \((1)(-1) -
(2)(-3) = -1 + 6 = 5\text{.}\) Therefore, I know that this matrix multiplies all areas by a factor of \(5\) and preserves orientation.
Subsection 8.2.2 The Cofactor Algorithm
Definition 8.2.2 .
The algorithm for calculating determinants is called Co-Factor Expansion . It is a recursive algorithm that reduces determinant calculations to determinants of smaller square matrices, eventually down to \(2 \times 2\) matrices whose determinants have already been defined.
Co-factor expansion proceeds in this way: I choose any column or row of the matrix. I take the coefficients from that column or row. For each of the coefficients, I multiply that coefficient by the determinant of the matrix formed by removing both the column and row containing that coefficient. Then I add up the determinants of these small matrices multiplied by matching coefficients, with a pattern of \(\pm\) signs. That pattern of \(\pm\) signs is a checkerboard pattern, as in the following \(5 \times 5\) matrix.
\begin{equation*}
\left( \begin{matrix} + \amp - \amp + \amp - \amp + \\ -
\amp + \amp - \amp + \amp - \\ + \amp - \amp + \amp -
\amp + \\ - \amp + \amp - \amp + \amp - \\ + \amp - \amp
+ \amp - \amp + \end{matrix} \right)
\end{equation*}
That’s a hard algorithm to intuit from the formal description. I’ll do an example.
Example 8.2.3 .
Here is a \(3 \times 3\) example where I choose the first row for co-factor expansion. For each element in the first row, I multiply the number by the determinant of the matrix with the first row and the matching column removed. I multiply by \((-1)\) in the second coefficient to match the checkerboard pattern of signs. After the first step, I can just calculate the three smaller determinants and simplify the arithmetic.
\begin{align*}
\amp \begin{vmatrix} 5 \amp -2 \amp 0 \\ -3 \amp 3 \amp -2
\\ 1 \amp -5 \amp 3 \end{vmatrix} = +5 \begin{vmatrix} 3
\amp -2 \\ -5 \amp 3 \end{vmatrix} -(-2) \begin{vmatrix}
-3 \amp -2 \\ 1 \amp 3 \end{vmatrix} +0 \begin{vmatrix} -3
\amp 3 \\ 1 \amp -5 \end{vmatrix} \\
\amp = 5((3)(3) - (-2)(-5)) + 2((-3)(3) - (-2)(1)) +
(0)((-3)(-5)-(3)(1))\\
\amp = 5(-1) + 2(-7) + 0 = -19
\end{align*}
If I did co-factor expansion of an arbitrary \(3 \times 3\) matrix, I would get a direct expression for the determinant.
\begin{equation*}
\left| \begin{matrix} a \amp b \amp c \\ d \amp e \amp f \\
g \amp h \amp i \end{matrix} \right| = aei - ahf - bdi + bfg +
cdh - ceg
\end{equation*}
I could use this formula for \(3 \times 3\) matrices if I wished. I could do the same for larger matrices, but it starts to get quite complicated. The computational complexity of the recursive determinant algorithm grows very quickly. For a \(3
\times 3\) matrix, I had 3 recursions to \(2 \times 2\) matrices, each with \(2\) multiplication terms in the determinant, giving \(6\) terms. Each term was the multiple of three coefficients, giving \(12\) total multiplications. For a \(4 \times 4\text{,}\) the recursion gives \(24\) terms, each with \(4\) coefficients, so \(24\cdot 3 = 72\) multiplications. For a \(5 \times 5\) matrix, I need to recurse five times to a \(4 \times 4\) matrix, for \(120\) terms each with \(5\) coefficients, which is \(120 \cdot 4 = 600\) multiplications. The pattern continues: there are \(n!\) terms in the determinant of a \(n \times n\) matrix, each with \(n\) coefficients, for \(n! (n-1)\) multiplications. This is computationally terrifying, making determinants of large matrices computationally very difficult.
