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Exercice

L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.


On considère la matrice $$ A= \begin{pmatrix}1 & 1 & \dfrac{7}{4} & 4 & 5 & -1 \\ -\dfrac{5}{4} & -1 & \dfrac{9}{2} & 3 & -5 & 1 \\ 5 & 5 & -4 & 3 & 3 & 1 \\ 4 & -\dfrac{13}{4} & -\dfrac{25}{4} & \dfrac{9}{4} & -\dfrac{23}{5} & -\dfrac{17}{4} \\ \dfrac{1}{2} & -4 & \dfrac{12}{5} & 0 & -\dfrac{1}{2} & \dfrac{11}{2} \\ -4 & -\dfrac{19}{5} & 0 & \dfrac{7}{2} & 3 & 3\end{pmatrix}$$
  1. Donner les mineurs d'ordre $ (5, 3)$ et $ (6, 2)$ : $ \widehat{A}_{5, 3}=$ $ \widehat{A}_{6, 2}=$
  2. Expliquer pourquoi $ \det(A)=\det \begin{pmatrix}1 & 1 & \dfrac{7}{4} & 4 & 5 & -1 \\ 0 & \dfrac{1}{4} & \dfrac{107}{16} & 8 & \dfrac{5}{4} & -\dfrac{1}{4} \\ 0 & 0 & -\dfrac{51}{4} & -17 & -22 & 6 \\ 0 & -\dfrac{29}{4} & -\dfrac{53}{4} & -\dfrac{55}{4} & -\dfrac{123}{5} & -\dfrac{1}{4} \\ 0 & -\dfrac{9}{2} & \dfrac{61}{40} & -2 & -3 & 6 \\ 0 & \dfrac{1}{5} & 7 & \dfrac{39}{2} & 23 & -1\end{pmatrix} $
  3. Calculer $ \det(A)$ .
  4. Pourquoi la matrice $ A $ est inversible.
  5. Donner $ B=A^{-1}$ l'inverse de la matrice $ A $ , en ne détaillant que le calcul de $ A^{-1}_{4, 2}$ .
  6. Donner $ B^{-1}$ l'inverse de la matrice $ B$ . Justifier.
  7. Résoudre le système suivant. $ \left\{\begin{array}{*{7}{cr}} &x&+&y &+&\dfrac{7}{4}z &+&4t &+&5u &-&v &=&0\\ &-\dfrac{5}{4}x&-&y &+&\dfrac{9}{2}z &+&3t &-&5u &+&v &=&8\\ &5x&+&5y &-&4z &+&3t &+&3u &+&v &=&8\\ &4x&-&\dfrac{13}{4}y &-&\dfrac{25}{4}z &+&\dfrac{9}{4}t &-&\dfrac{23}{5}u &-&\dfrac{17}{4}v &=&-9\\ &\dfrac{1}{2}x&-&4y &+&\dfrac{12}{5}z &&&-&\dfrac{1}{2}u &+&\dfrac{11}{2}v &=&7\\ &-4x&-&\dfrac{19}{5}y &&&+&\dfrac{7}{2}t &+&3u &+&3v &=&-9\\ \end{array} \right. $
  8. Résoudre le système suivant. $ \left\{\begin{array}{*{7}{cr}} &\dfrac{1.9752586051584E+35}{1.9352635493581E+36}x&-&\dfrac{1.1762924309709E+36}{3.8705270987162E+37}y &+&\dfrac{4.5840542466048E+35}{3.8705270987162E+37}z &+&\dfrac{3.04614998016E+31}{6.047698591744E+32}t &+&\dfrac{3.228680323072E+32}{2.4190794366976E+33}u &-&\dfrac{1.610255695872E+32}{1.2095397183488E+33}v &=&\dfrac{55}{3}\\ &-\dfrac{7.02844239872E+31}{9.6763177467904E+32}x&+&\dfrac{5.97123661824E+31}{9.6763177467904E+32}y &+&\dfrac{2.3907923709133E+38}{2.4771373431783E+39}z &-&\dfrac{1.455816704E+31}{1.9352635493581E+32}t &-&\dfrac{7.5696439296E+31}{7.7410541974323E+32}u &-&\dfrac{1.64468621312E+31}{3.8705270987162E+33}v &=&-6\\ &\dfrac{2.255488548864E+32}{1.9352635493581E+33}x&+&\dfrac{2.7393523712E+30}{6.047698591744E+31}y &-&\dfrac{1.282144731136E+32}{1.9352635493581E+33}z &-&\dfrac{3.959947264E+29}{1.2095397183488E+31}t &+&\dfrac{2.47118430208E+31}{4.8381588733952E+32}u &-&\dfrac{1.82277636096E+31}{1.9352635493581E+32}v &=&9\\ &\dfrac{5.13361313792E+31}{1.2095397183488E+33}x&+&\dfrac{1.9609848853299E+38}{1.9352635493581E+39}y &+&\dfrac{3.6894652560835E+37}{6.1928433579459E+38}z &+&\dfrac{7.50649868288E+31}{2.4190794366976E+33}t &-&\dfrac{1.975794532352E+32}{4.8381588733952E+33}u &+&\dfrac{1.5354642825216E+33}{1.9352635493581E+34}v &=&2\\ &\dfrac{1.907552157696E+32}{1.9352635493581E+33}x&-&\dfrac{9.619479461888E+32}{9.6763177467904E+33}y &-&\dfrac{1.0462377279488E+33}{3.8705270987162E+34}z &-&\dfrac{6.8573200384E+30}{3.8705270987162E+32}t &+&\dfrac{1.12909090816E+31}{4.8381588733952E+32}u &+&\dfrac{1.7224433664E+30}{2.4190794366976E+32}v &=&1\\ &-\dfrac{8.0507488436224E+41}{7.7410541974323E+42}x&+&\dfrac{1.166715249492E+45}{6.1928433579459E+46}y &+&\dfrac{1.893881060199E+46}{1.9817098745427E+47}z &-&\dfrac{4.510421876736E+32}{9.6763177467904E+33}t &+&\dfrac{1.5172728520704E+33}{1.9352635493581E+34}u &+&\dfrac{1.227915526144E+32}{2.4190794366976E+33}v &=&9\\ \end{array} \right. $
Cliquer ici pour afficher la solution
  1. $ \widehat{A}_{5, 3}=\begin{pmatrix}1 & 1 & 4 & 5 & -1 \\ -\dfrac{5}{4} & -1 & 3 & -5 & 1 \\ 5 & 5 & 3 & 3 & 1 \\ 4 & -\dfrac{13}{4} & \dfrac{9}{4} & -\dfrac{23}{5} & -\dfrac{17}{4} \\ -4 & -\dfrac{19}{5} & \dfrac{7}{2} & 3 & 3\end{pmatrix}$ $ \widehat{A}_{6, 2}=\begin{pmatrix}1 & \dfrac{7}{4} & 4 & 5 & -1 \\ -\dfrac{5}{4} & \dfrac{9}{2} & 3 & -5 & 1 \\ 5 & -4 & 3 & 3 & 1 \\ 4 & -\dfrac{25}{4} & \dfrac{9}{4} & -\dfrac{23}{5} & -\dfrac{17}{4} \\ \dfrac{1}{2} & \dfrac{12}{5} & 0 & -\dfrac{1}{2} & \dfrac{11}{2}\end{pmatrix}$
  2. On sait, d'après le cours, que l'on ne modifie la valeur du déterminant d'une matrice lorsqu'on ajoute à une ligne une combinaison linéaire des autres. On est donc partie de la matrice $ A$ et on a fait : $ L_{2}\leftarrow L_{2}-\left(-\dfrac{5}{4}\right)L_1$ , $ L_{3}\leftarrow L_{3}-\left(5\right)L_1$ , $ L_{4}\leftarrow L_{4}-\left(4\right)L_1$ , $ L_{5}\leftarrow L_{5}-\left(\dfrac{1}{2}\right)L_1$ et $ L_{6}\leftarrow L_{6}-\left(-4\right)L_1$
  3. En développant par rapport à la première colonne, en se servant de la précédente remarque on a \begin{eqnarray*} \det(A) &=&\det\begin{pmatrix}1 & 1 & \dfrac{7}{4} & 4 & 5 & -1 \\ 0 & \dfrac{1}{4} & \dfrac{107}{16} & 8 & \dfrac{5}{4} & -\dfrac{1}{4} \\ 0 & 0 & -\dfrac{51}{4} & -17 & -22 & 6 \\ 0 & -\dfrac{29}{4} & -\dfrac{53}{4} & -\dfrac{55}{4} & -\dfrac{123}{5} & -\dfrac{1}{4} \\ 0 & -\dfrac{9}{2} & \dfrac{61}{40} & -2 & -3 & 6 \\ 0 & \dfrac{1}{5} & 7 & \dfrac{39}{2} & 23 & -1\end{pmatrix}\\ &=&1\times\det\begin{pmatrix}\dfrac{1}{4} & \dfrac{107}{16} & 8 & \dfrac{5}{4} & -\dfrac{1}{4} \\ 0 & -\dfrac{51}{4} & -17 & -22 & 6 \\ -\dfrac{29}{4} & -\dfrac{53}{4} & -\dfrac{55}{4} & -\dfrac{123}{5} & -\dfrac{1}{4} \\ -\dfrac{9}{2} & \dfrac{61}{40} & -2 & -3 & 6 \\ \dfrac{1}{5} & 7 & \dfrac{39}{2} & 23 & -1\end{pmatrix}\\ &=&\dfrac{6.047698591744E+30}{5.24288E+25} \end{eqnarray*}
  4. On observe que le déterminant de $ A$ est non nul. D'après le cours, cela signifie que la matrice est inversible.
  5. D'après le cours $ B=A^{-1}=\left(\dfrac{6.047698591744E+30}{5.24288E+25}\right)^{-1}{}^tCo\begin{pmatrix}1 & 1 & \dfrac{7}{4} & 4 & 5 & -1 \\ -\dfrac{5}{4} & -1 & \dfrac{9}{2} & 3 & -5 & 1 \\ 5 & 5 & -4 & 3 & 3 & 1 \\ 4 & -\dfrac{13}{4} & -\dfrac{25}{4} & \dfrac{9}{4} & -\dfrac{23}{5} & -\dfrac{17}{4} \\ \dfrac{1}{2} & -4 & \dfrac{12}{5} & 0 & -\dfrac{1}{2} & \dfrac{11}{2} \\ -4 & -\dfrac{19}{5} & 0 & \dfrac{7}{2} & 3 & 3\end{pmatrix} =\dfrac{5.24288E+25}{6.047698591744E+30}\begin{pmatrix}\dfrac{1.1945768684747E+66}{1.0146354557658E+62} & -\dfrac{7.1138620782617E+66}{2.0292709115317E+63} & \dfrac{2.772297841167E+66}{2.0292709115317E+63} & \dfrac{1.8422196945255E+62}{3.1707357992683E+58} & \dfrac{1.9526085443034E+63}{1.2682943197073E+59} & -\dfrac{9.7383411042728E+62}{6.3414715985366E+58} \\ -\dfrac{4.2505901196893E+62}{5.0731772788292E+58} & \dfrac{3.61122392871E+62}{5.0731772788292E+58} & \dfrac{1.4458791654725E+69}{1.2987333833803E+65} & -\dfrac{8.8043406306182E+61}{1.0146354557658E+58} & -\dfrac{4.5778924933045E+62}{4.0585418230634E+58} & -\dfrac{9.9465664949466E+61}{2.0292709115317E+59} \\ \dfrac{1.364051492066E+63}{1.0146354557658E+59} & \dfrac{1.6566777477597E+61}{3.1707357992683E+57} & -\dfrac{7.7540248849032E+62}{1.0146354557658E+59} & -\dfrac{2.3948567491873E+60}{6.3414715985366E+56} & \dfrac{1.4944977823629E+62}{2.5365886394146E+58} & -\dfrac{1.1023602031242E+62}{1.0146354557658E+58} \\ \dfrac{3.1046544944757E+62}{6.3414715985366E+58} & \dfrac{1.1859445529441E+69}{1.0146354557658E+65} & \dfrac{2.2312773833504E+68}{3.2468334584507E+64} & \dfrac{4.5397041513382E+62}{1.2682943197073E+59} & -\dfrac{1.1949009810881E+63}{2.5365886394146E+59} & \dfrac{9.2860251790791E+63}{1.0146354557658E+60} \\ \dfrac{1.1536300497776E+63}{1.0146354557658E+59} & -\dfrac{5.817571239497E+63}{5.0731772788292E+59} & -\dfrac{6.3273304339454E+63}{2.0292709115317E+60} & -\dfrac{4.147100473937E+61}{2.0292709115317E+58} & \dfrac{6.8284014952302E+61}{2.5365886394146E+58} & \dfrac{1.0416818321336E+61}{1.2682943197073E+58} \\ -\dfrac{4.868850244406E+72}{4.0585418230634E+68} & \dfrac{7.0559421713188E+75}{3.2468334584507E+72} & \dfrac{1.1453621820696E+77}{1.0389867067042E+73} & -\dfrac{2.7277672032108E+63}{5.0731772788292E+59} & \dfrac{9.1760088907576E+63}{1.0146354557658E+60} & \dfrac{7.4260629982417E+62}{1.2682943197073E+59}\end{pmatrix} =\begin{pmatrix}\dfrac{1.9752586051584E+35}{1.9352635493581E+36} & -\dfrac{1.1762924309709E+36}{3.8705270987162E+37} & \dfrac{4.5840542466048E+35}{3.8705270987162E+37} & \dfrac{3.04614998016E+31}{6.047698591744E+32} & \dfrac{3.228680323072E+32}{2.4190794366976E+33} & -\dfrac{1.610255695872E+32}{1.2095397183488E+33} \\ -\dfrac{7.02844239872E+31}{9.6763177467904E+32} & \dfrac{5.97123661824E+31}{9.6763177467904E+32} & \dfrac{2.3907923709133E+38}{2.4771373431783E+39} & -\dfrac{1.455816704E+31}{1.9352635493581E+32} & -\dfrac{7.5696439296E+31}{7.7410541974323E+32} & -\dfrac{1.64468621312E+31}{3.8705270987162E+33} \\ \dfrac{2.255488548864E+32}{1.9352635493581E+33} & \dfrac{2.7393523712E+30}{6.047698591744E+31} & -\dfrac{1.282144731136E+32}{1.9352635493581E+33} & -\dfrac{3.959947264E+29}{1.2095397183488E+31} & \dfrac{2.47118430208E+31}{4.8381588733952E+32} & -\dfrac{1.82277636096E+31}{1.9352635493581E+32} \\ \dfrac{5.13361313792E+31}{1.2095397183488E+33} & \dfrac{1.9609848853299E+38}{1.9352635493581E+39} & \dfrac{3.6894652560835E+37}{6.1928433579459E+38} & \dfrac{7.50649868288E+31}{2.4190794366976E+33} & -\dfrac{1.975794532352E+32}{4.8381588733952E+33} & \dfrac{1.5354642825216E+33}{1.9352635493581E+34} \\ \dfrac{1.907552157696E+32}{1.9352635493581E+33} & -\dfrac{9.619479461888E+32}{9.6763177467904E+33} & -\dfrac{1.0462377279488E+33}{3.8705270987162E+34} & -\dfrac{6.8573200384E+30}{3.8705270987162E+32} & \dfrac{1.12909090816E+31}{4.8381588733952E+32} & \dfrac{1.7224433664E+30}{2.4190794366976E+32} \\ -\dfrac{8.0507488436224E+41}{7.7410541974323E+42} & \dfrac{1.166715249492E+45}{6.1928433579459E+46} & \dfrac{1.893881060199E+46}{1.9817098745427E+47} & -\dfrac{4.510421876736E+32}{9.6763177467904E+33} & \dfrac{1.5172728520704E+33}{1.9352635493581E+34} & \dfrac{1.227915526144E+32}{2.4190794366976E+33}\end{pmatrix}$ . Précisément on a calculé $ A^{-1}_{4, 2}=B_{4, 2}= \left(\dfrac{6.047698591744E+30}{5.24288E+25}\right)^{-1}Co(A)_{2, 4}= \left(\dfrac{6.047698591744E+30}{5.24288E+25}\right)^{-1}\times(-1)^{2+4}\det\begin{pmatrix}1 & 1 & \dfrac{7}{4} & 5 & -1 \\ 5 & 5 & -4 & 3 & 1 \\ 4 & -\dfrac{13}{4} & -\dfrac{25}{4} & -\dfrac{23}{5} & -\dfrac{17}{4} \\ \dfrac{1}{2} & -4 & \dfrac{12}{5} & -\dfrac{1}{2} & \dfrac{11}{2} \\ -4 & -\dfrac{19}{5} & 0 & 3 & 3\end{pmatrix}=\dfrac{1.9609848853299E+38}{1.9352635493581E+39}$ .
  6. On observe que $ B^{-1}=\left(A^{-1}\right)^{-1}=A$ . Trivialement, et sans calcul, $$ B^{-1}=\begin{pmatrix}1 & 1 & \dfrac{7}{4} & 4 & 5 & -1 \\ -\dfrac{5}{4} & -1 & \dfrac{9}{2} & 3 & -5 & 1 \\ 5 & 5 & -4 & 3 & 3 & 1 \\ 4 & -\dfrac{13}{4} & -\dfrac{25}{4} & \dfrac{9}{4} & -\dfrac{23}{5} & -\dfrac{17}{4} \\ \dfrac{1}{2} & -4 & \dfrac{12}{5} & 0 & -\dfrac{1}{2} & \dfrac{11}{2} \\ -4 & -\dfrac{19}{5} & 0 & \dfrac{7}{2} & 3 & 3\end{pmatrix}$$
  7. Le système $ \left\{\begin{array}{*{7}{cr}} &x&+&y &+&\dfrac{7}{4}z &+&4t &+&5u &-&v &=&0\\ &-\dfrac{5}{4}x&-&y &+&\dfrac{9}{2}z &+&3t &-&5u &+&v &=&8\\ &5x&+&5y &-&4z &+&3t &+&3u &+&v &=&8\\ &4x&-&\dfrac{13}{4}y &-&\dfrac{25}{4}z &+&\dfrac{9}{4}t &-&\dfrac{23}{5}u &-&\dfrac{17}{4}v &=&-9\\ &\dfrac{1}{2}x&-&4y &+&\dfrac{12}{5}z &&&-&\dfrac{1}{2}u &+&\dfrac{11}{2}v &=&7\\ &-4x&-&\dfrac{19}{5}y &&&+&\dfrac{7}{2}t &+&3u &+&3v &=&-9\\ \end{array} \right.$ est équivalent à l'équation $ AX=a$ où la matrice $ A$ est celle de l'énoncé, $ X$ la matrice colonne des indéterminés et $ a=\begin{pmatrix}0 \\ 8 \\ 8 \\ -9 \\ 7 \\ -9\end{pmatrix}$ de sorte que la solution est $ X=A^{-1}a=\begin{pmatrix}\dfrac{1.9752586051584E+35}{1.9352635493581E+36} & -\dfrac{1.1762924309709E+36}{3.8705270987162E+37} & \dfrac{4.5840542466048E+35}{3.8705270987162E+37} & \dfrac{3.04614998016E+31}{6.047698591744E+32} & \dfrac{3.228680323072E+32}{2.4190794366976E+33} & -\dfrac{1.610255695872E+32}{1.2095397183488E+33} \\ -\dfrac{7.02844239872E+31}{9.6763177467904E+32} & \dfrac{5.97123661824E+31}{9.6763177467904E+32} & \dfrac{2.3907923709133E+38}{2.4771373431783E+39} & -\dfrac{1.455816704E+31}{1.9352635493581E+32} & -\dfrac{7.5696439296E+31}{7.7410541974323E+32} & -\dfrac{1.64468621312E+31}{3.8705270987162E+33} \\ \dfrac{2.255488548864E+32}{1.9352635493581E+33} & \dfrac{2.7393523712E+30}{6.047698591744E+31} & -\dfrac{1.282144731136E+32}{1.9352635493581E+33} & -\dfrac{3.959947264E+29}{1.2095397183488E+31} & \dfrac{2.47118430208E+31}{4.8381588733952E+32} & -\dfrac{1.82277636096E+31}{1.9352635493581E+32} \\ \dfrac{5.13361313792E+31}{1.2095397183488E+33} & \dfrac{1.9609848853299E+38}{1.9352635493581E+39} & \dfrac{3.6894652560835E+37}{6.1928433579459E+38} & \dfrac{7.50649868288E+31}{2.4190794366976E+33} & -\dfrac{1.975794532352E+32}{4.8381588733952E+33} & \dfrac{1.5354642825216E+33}{1.9352635493581E+34} \\ \dfrac{1.907552157696E+32}{1.9352635493581E+33} & -\dfrac{9.619479461888E+32}{9.6763177467904E+33} & -\dfrac{1.0462377279488E+33}{3.8705270987162E+34} & -\dfrac{6.8573200384E+30}{3.8705270987162E+32} & \dfrac{1.12909090816E+31}{4.8381588733952E+32} & \dfrac{1.7224433664E+30}{2.4190794366976E+32} \\ -\dfrac{8.0507488436224E+41}{7.7410541974323E+42} & \dfrac{1.166715249492E+45}{6.1928433579459E+46} & \dfrac{1.893881060199E+46}{1.9817098745427E+47} & -\dfrac{4.510421876736E+32}{9.6763177467904E+33} & \dfrac{1.5172728520704E+33}{1.9352635493581E+34} & \dfrac{1.227915526144E+32}{2.4190794366976E+33}\end{pmatrix}\times\begin{pmatrix}0 \\ 8 \\ 8 \\ -9 \\ 7 \\ -9\end{pmatrix}=\begin{pmatrix}\dfrac{4.0579037759868E+174}{2.6509444494364E+174} \\ \dfrac{1.802046256835E+171}{1.3898583635061E+171} \\ \dfrac{1.7658372859281E+161}{1.3254722247182E+161} \\ \dfrac{2.1806148079323E+177}{2.7145671162228E+179} \\ -\dfrac{1.2772358019659E+166}{1.6966044476393E+166} \\ \dfrac{7.9319328181198E+195}{5.5594334540243E+195}\end{pmatrix}$ . Ainsi $ x=\dfrac{4.0579037759868E+174}{2.6509444494364E+174}$ , $ y=\dfrac{1.802046256835E+171}{1.3898583635061E+171}$ , $ z=\dfrac{1.7658372859281E+161}{1.3254722247182E+161}$ , $ t=\dfrac{2.1806148079323E+177}{2.7145671162228E+179}$ , $ u=\dfrac{1.2772358019659E+166}{1.6966044476393E+166}$ et $ v=\dfrac{7.9319328181198E+195}{5.5594334540243E+195}$
  8. Le système $ \left\{\begin{array}{*{7}{cr}} &\dfrac{1.9752586051584E+35}{1.9352635493581E+36}x&-&\dfrac{1.1762924309709E+36}{3.8705270987162E+37}y &+&\dfrac{4.5840542466048E+35}{3.8705270987162E+37}z &+&\dfrac{3.04614998016E+31}{6.047698591744E+32}t &+&\dfrac{3.228680323072E+32}{2.4190794366976E+33}u &-&\dfrac{1.610255695872E+32}{1.2095397183488E+33}v &=&\dfrac{55}{3}\\ &-\dfrac{7.02844239872E+31}{9.6763177467904E+32}x&+&\dfrac{5.97123661824E+31}{9.6763177467904E+32}y &+&\dfrac{2.3907923709133E+38}{2.4771373431783E+39}z &-&\dfrac{1.455816704E+31}{1.9352635493581E+32}t &-&\dfrac{7.5696439296E+31}{7.7410541974323E+32}u &-&\dfrac{1.64468621312E+31}{3.8705270987162E+33}v &=&-6\\ &\dfrac{2.255488548864E+32}{1.9352635493581E+33}x&+&\dfrac{2.7393523712E+30}{6.047698591744E+31}y &-&\dfrac{1.282144731136E+32}{1.9352635493581E+33}z &-&\dfrac{3.959947264E+29}{1.2095397183488E+31}t &+&\dfrac{2.47118430208E+31}{4.8381588733952E+32}u &-&\dfrac{1.82277636096E+31}{1.9352635493581E+32}v &=&9\\ &\dfrac{5.13361313792E+31}{1.2095397183488E+33}x&+&\dfrac{1.9609848853299E+38}{1.9352635493581E+39}y &+&\dfrac{3.6894652560835E+37}{6.1928433579459E+38}z &+&\dfrac{7.50649868288E+31}{2.4190794366976E+33}t &-&\dfrac{1.975794532352E+32}{4.8381588733952E+33}u &+&\dfrac{1.5354642825216E+33}{1.9352635493581E+34}v &=&2\\ &\dfrac{1.907552157696E+32}{1.9352635493581E+33}x&-&\dfrac{9.619479461888E+32}{9.6763177467904E+33}y &-&\dfrac{1.0462377279488E+33}{3.8705270987162E+34}z &-&\dfrac{6.8573200384E+30}{3.8705270987162E+32}t &+&\dfrac{1.12909090816E+31}{4.8381588733952E+32}u &+&\dfrac{1.7224433664E+30}{2.4190794366976E+32}v &=&1\\ &-\dfrac{8.0507488436224E+41}{7.7410541974323E+42}x&+&\dfrac{1.166715249492E+45}{6.1928433579459E+46}y &+&\dfrac{1.893881060199E+46}{1.9817098745427E+47}z &-&\dfrac{4.510421876736E+32}{9.6763177467904E+33}t &+&\dfrac{1.5172728520704E+33}{1.9352635493581E+34}u &+&\dfrac{1.227915526144E+32}{2.4190794366976E+33}v &=&9\\ \end{array} \right.$ est équivalent à l'équation $ BX=b$ où la matrice $ B=A^{-1}$ déterminée précédement, $ X$ la matrice colonne des indéterminés et $ b=\begin{pmatrix}\dfrac{55}{3} \\ -6 \\ 9 \\ 2 \\ 1 \\ 9\end{pmatrix}$ de sorte que la solution est $ X=B^{-1}b=Ab=\begin{pmatrix}1 & 1 & \dfrac{7}{4} & 4 & 5 & -1 \\ -\dfrac{5}{4} & -1 & \dfrac{9}{2} & 3 & -5 & 1 \\ 5 & 5 & -4 & 3 & 3 & 1 \\ 4 & -\dfrac{13}{4} & -\dfrac{25}{4} & \dfrac{9}{4} & -\dfrac{23}{5} & -\dfrac{17}{4} \\ \dfrac{1}{2} & -4 & \dfrac{12}{5} & 0 & -\dfrac{1}{2} & \dfrac{11}{2} \\ -4 & -\dfrac{19}{5} & 0 & \dfrac{7}{2} & 3 & 3\end{pmatrix}\times\begin{pmatrix}\dfrac{55}{3} \\ -6 \\ 9 \\ 2 \\ 1 \\ 9\end{pmatrix}=\begin{pmatrix}\dfrac{385}{12} \\ \dfrac{403}{12} \\ \dfrac{131}{3} \\ -\dfrac{53}{30} \\ \dfrac{3113}{30} \\ -\dfrac{203}{15}\end{pmatrix}$ . Ainsi $ x=\dfrac{385}{12}$ , $ y=\dfrac{403}{12}$ , $ z=\dfrac{131}{3}$ , $ t=\dfrac{53}{30}$ , $ u=\dfrac{3113}{30}$ et $ v=\dfrac{203}{15}$