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.
Exercice
On considère la matrice
\[ A= \begin{pmatrix}1 & -3 & -1 & -\dfrac{1}{2} & 3 & 1 \\ -\dfrac{1}{4} & 2 & -\dfrac{14}{5} & 0 & -4 & -1 \\ -3 & 4 & 3 & \dfrac{11}{4} & \dfrac{25}{2} & -\dfrac{17}{5} \\ \dfrac{5}{4} & 2 & -3 & 2 & 4 & 1 \\ \dfrac{4}{3} & \dfrac{12}{5} & -4 & \dfrac{5}{3} & 3 & \dfrac{22}{3} \\ \dfrac{7}{5} & -\dfrac{21}{4} & -4 & -4 & -5 & -5\end{pmatrix}\]
- Donner les mineurs d'ordre \( (6, 2)\) et \( (5, 4)\) :
\( \widehat{A}_{6, 2}=\)
\( \widehat{A}_{5, 4}=\)
- Expliquer pourquoi
\( \det(A)=\det
\begin{pmatrix}1 & -3 & -1 & -\dfrac{1}{2} & 3 & 1 \\ 0 & \dfrac{5}{4} & -\dfrac{61}{20} & -\dfrac{1}{8} & -\dfrac{13}{4} & -\dfrac{3}{4} \\ 0 & -5 & 0 & \dfrac{5}{4} & \dfrac{43}{2} & -\dfrac{2}{5} \\ 0 & \dfrac{23}{4} & -\dfrac{7}{4} & \dfrac{21}{8} & \dfrac{1}{4} & -\dfrac{1}{4} \\ 0 & \dfrac{32}{5} & -\dfrac{8}{3} & \dfrac{7}{3} & -1 & 6 \\ 0 & -\dfrac{21}{20} & -\dfrac{13}{5} & -\dfrac{33}{10} & -\dfrac{46}{5} & -\dfrac{32}{5}\end{pmatrix}
\)
- Calculer \( \det(A)\) .
- Pourquoi la matrice \( A \) est inversible.
- Donner \( B=A^{-1}\) l'inverse de la matrice \( A \) , en ne détaillant que le calcul de \( A^{-1}_{5, 1}\) .
- Donner \( B^{-1}\) l'inverse de la matrice \( B\) . Justifier.
- Résoudre le système suivant.
\( \left\{\begin{array}{*{7}{cr}}
&x&-&3y &-&z &-&\dfrac{1}{2}t &+&3u &+&v &=&-6\\
&-\dfrac{1}{4}x&+&2y &-&\dfrac{14}{5}z &&&-&4u &-&v &=&2\\
&-3x&+&4y &+&3z &+&\dfrac{11}{4}t &+&\dfrac{25}{2}u &-&\dfrac{17}{5}v &=&0\\
&\dfrac{5}{4}x&+&2y &-&3z &+&2t &+&4u &+&v &=&6\\
&\dfrac{4}{3}x&+&\dfrac{12}{5}y &-&4z &+&\dfrac{5}{3}t &+&3u &+&\dfrac{22}{3}v &=&-5\\
&\dfrac{7}{5}x&-&\dfrac{21}{4}y &-&4z &-&4t &-&5u &-&5v &=&9\\
\end{array}
\right.
\)
- Résoudre le système suivant.
\( \left\{\begin{array}{*{7}{cr}}
&-\dfrac{1.6662645504E+23}{1.54916892288E+23}x&-&\dfrac{4.942165248E+22}{4.64750676864E+22}y &-&\dfrac{6.74572032E+21}{4.64750676864E+22}z &+&\dfrac{8.920689408E+22}{1.54916892288E+23}t &+&\dfrac{6.559488E+19}{6.19667569152E+20}u &+&\dfrac{4.260420864E+22}{1.16187669216E+23}v &=&7\\
&-\dfrac{1.27486192896E+26}{1.394252030592E+26}x&-&\dfrac{5.416342272E+22}{9.29501353728E+22}y &+&\dfrac{2.94425856E+21}{4.64750676864E+22}z &+&\dfrac{1.550018304E+22}{1.54916892288E+23}t &+&\dfrac{4.386016512E+22}{1.54916892288E+23}u &+&\dfrac{3.028730112E+22}{9.29501353728E+22}v &=&-4\\
&-\dfrac{5.688850176E+22}{1.54916892288E+23}x&-&\dfrac{7.5010597632E+25}{1.859002707456E+26}y &-&\dfrac{4.7884032E+20}{1.16187669216E+22}z &+&\dfrac{3.732505344E+22}{6.19667569152E+23}t &+&\dfrac{1.057536E+19}{3.09833784576E+22}u &+&\dfrac{5.5570176E+20}{1.16187669216E+22}v &=&6\\
&\dfrac{6.301225728E+22}{4.841152884E+22}x&+&\dfrac{9.545819904E+22}{9.29501353728E+22}y &-&\dfrac{8.070204672E+22}{4.64750676864E+23}z &+&\dfrac{1.4457648384E+23}{7.7458446144E+23}t &-&\dfrac{1.9153269504E+23}{3.09833784576E+23}u &-&\dfrac{4.045250304E+22}{5.8093834608E+22}v &=&9\\
&-\dfrac{2.2878713088E+23}{1.54916892288E+24}x&-&\dfrac{1.8128155392E+23}{9.29501353728E+23}y &+&\dfrac{1.6549632E+20}{2.32375338432E+21}z &+&\dfrac{1.488538368E+22}{1.54916892288E+24}t &+&\dfrac{1.289689344E+22}{1.239335138304E+23}u &+&\dfrac{2.685973248E+22}{2.32375338432E+23}v &=&-\dfrac{36}{7}\\
&\dfrac{3.662648064E+22}{6.19667569152E+23}x&+&\dfrac{1.28173824E+21}{1.239335138304E+23}y &-&\dfrac{2.01164544E+21}{3.09833784576E+23}z &-&\dfrac{9.353219328E+22}{6.19667569152E+23}t &+&\dfrac{1.51905024E+21}{1.239335138304E+22}u &-&\dfrac{1.12317696E+21}{3.09833784576E+22}v &=&-9\\
\end{array}
\right.
\)
Cliquer ici pour afficher la solution
Exercice
-
\( \widehat{A}_{6, 2}=\begin{pmatrix}1 & -1 & -\dfrac{1}{2} & 3 & 1 \\ -\dfrac{1}{4} & -\dfrac{14}{5} & 0 & -4 & -1 \\ -3 & 3 & \dfrac{11}{4} & \dfrac{25}{2} & -\dfrac{17}{5} \\ \dfrac{5}{4} & -3 & 2 & 4 & 1 \\ \dfrac{4}{3} & -4 & \dfrac{5}{3} & 3 & \dfrac{22}{3}\end{pmatrix}\)
\( \widehat{A}_{5, 4}=\begin{pmatrix}1 & -3 & -1 & 3 & 1 \\ -\dfrac{1}{4} & 2 & -\dfrac{14}{5} & -4 & -1 \\ -3 & 4 & 3 & \dfrac{25}{2} & -\dfrac{17}{5} \\ \dfrac{5}{4} & 2 & -3 & 4 & 1 \\ \dfrac{7}{5} & -\dfrac{21}{4} & -4 & -5 & -5\end{pmatrix}\)
- 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 : \( L_{2}\leftarrow L_{2}-\left(-\dfrac{1}{4}\right)L_1\) , \( L_{3}\leftarrow L_{3}-\left(-3\right)L_1\) , \( L_{4}\leftarrow L_{4}-\left(\dfrac{5}{4}\right)L_1\) , \( L_{5}\leftarrow L_{5}-\left(\dfrac{4}{3}\right)L_1\) et \( L_{6}\leftarrow L_{6}-\left(\dfrac{7}{5}\right)L_1\)
- 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 & -3 & -1 & -\dfrac{1}{2} & 3 & 1 \\ 0 & \dfrac{5}{4} & -\dfrac{61}{20} & -\dfrac{1}{8} & -\dfrac{13}{4} & -\dfrac{3}{4} \\ 0 & -5 & 0 & \dfrac{5}{4} & \dfrac{43}{2} & -\dfrac{2}{5} \\ 0 & \dfrac{23}{4} & -\dfrac{7}{4} & \dfrac{21}{8} & \dfrac{1}{4} & -\dfrac{1}{4} \\ 0 & \dfrac{32}{5} & -\dfrac{8}{3} & \dfrac{7}{3} & -1 & 6 \\ 0 & -\dfrac{21}{20} & -\dfrac{13}{5} & -\dfrac{33}{10} & -\dfrac{46}{5} & -\dfrac{32}{5}\end{pmatrix}\\
&=&1\times\det\begin{pmatrix}\dfrac{5}{4} & -\dfrac{61}{20} & -\dfrac{1}{8} & -\dfrac{13}{4} & -\dfrac{3}{4} \\ -5 & 0 & \dfrac{5}{4} & \dfrac{43}{2} & -\dfrac{2}{5} \\ \dfrac{23}{4} & -\dfrac{7}{4} & \dfrac{21}{8} & \dfrac{1}{4} & -\dfrac{1}{4} \\ \dfrac{32}{5} & -\dfrac{8}{3} & \dfrac{7}{3} & -1 & 6 \\ -\dfrac{21}{20} & -\dfrac{13}{5} & -\dfrac{33}{10} & -\dfrac{46}{5} & -\dfrac{32}{5}\end{pmatrix}\\
&=&\dfrac{7.7458446144E+19}{23040000000000000}
\end{eqnarray*}
- On observe que le déterminant de \( A\) est non nul. D'après le cours, cela signifie que la matrice est inversible.
- D'après le cours
\(
B=A^{-1}=\left(\dfrac{7.7458446144E+19}{23040000000000000}\right)^{-1}{}^tCo\begin{pmatrix}1 & -3 & -1 & -\dfrac{1}{2} & 3 & 1 \\ -\dfrac{1}{4} & 2 & -\dfrac{14}{5} & 0 & -4 & -1 \\ -3 & 4 & 3 & \dfrac{11}{4} & \dfrac{25}{2} & -\dfrac{17}{5} \\ \dfrac{5}{4} & 2 & -3 & 2 & 4 & 1 \\ \dfrac{4}{3} & \dfrac{12}{5} & -4 & \dfrac{5}{3} & 3 & \dfrac{22}{3} \\ \dfrac{7}{5} & -\dfrac{21}{4} & -4 & -4 & -5 & -5\end{pmatrix}
=\dfrac{23040000000000000}{7.7458446144E+19}\begin{pmatrix}-\dfrac{1.2906626293881E+43}{3.5692851983155E+39} & -\dfrac{3.8281244069696E+42}{1.0707855594947E+39} & -\dfrac{5.2251301410921E+41}{1.0707855594947E+39} & \dfrac{6.9098274007692E+42}{3.5692851983155E+39} & \dfrac{5.0808774798021E+39}{1.4277140793262E+37} & \dfrac{3.3000558004492E+42}{2.6769638987366E+39} \\ -\dfrac{9.8748824065384E+45}{3.212356678484E+42} & -\dfrac{4.1954145617318E+42}{2.1415711189893E+39} & \dfrac{2.2805769310377E+41}{1.0707855594947E+39} & \dfrac{1.200620093226E+42}{3.5692851983155E+39} & \dfrac{3.3973402378145E+42}{3.5692851983155E+39} & \dfrac{2.3460072826506E+42}{2.1415711189893E+39} \\ -\dfrac{4.4064949497898E+42}{3.5692851983155E+39} & -\dfrac{5.8102043369075E+45}{4.2831422379786E+42} & -\dfrac{3.7090227138296E+40}{2.6769638987366E+38} & \dfrac{2.8911406417042E+42}{1.4277140793262E+40} & \dfrac{8.1915095301341E+38}{7.138570396631E+38} & \dfrac{4.3043794849086E+40}{2.6769638987366E+38} \\ \dfrac{4.8808315369348E+42}{1.1154016244736E+39} & \dfrac{7.3940437693431E+42}{2.1415711189893E+39} & -\dfrac{6.2510551395717E+42}{1.0707855594947E+40} & \dfrac{1.119866978721E+43}{1.7846425991578E+40} & -\dfrac{1.4835824943571E+43}{7.138570396631E+39} & -\dfrac{3.1333880281138E+42}{1.3384819493683E+39} \\ -\dfrac{1.7721495655709E+43}{3.5692851983155E+40} & -\dfrac{1.4041787481213E+43}{2.1415711189893E+40} & \dfrac{1.281908778975E+40}{5.3539277974733E+37} & \dfrac{1.1529986901101E+42}{3.5692851983155E+40} & \dfrac{9.9897332594715E+41}{2.8554281586524E+39} & \dfrac{2.0805131417443E+42}{5.3539277974733E+39} \\ \dfrac{2.8370302780977E+42}{1.4277140793262E+40} & \dfrac{9.9281452433745E+40}{2.8554281586524E+39} & -\dfrac{1.5581892997506E+41}{7.138570396631E+39} & -\dfrac{7.2448583559091E+42}{1.4277140793262E+40} & \dfrac{1.1766327120507E+41}{2.8554281586524E+38} & -\dfrac{8.6999542066342E+40}{7.138570396631E+38}\end{pmatrix}
=\begin{pmatrix}-\dfrac{1.6662645504E+23}{1.54916892288E+23} & -\dfrac{4.942165248E+22}{4.64750676864E+22} & -\dfrac{6.74572032E+21}{4.64750676864E+22} & \dfrac{8.920689408E+22}{1.54916892288E+23} & \dfrac{6.559488E+19}{6.19667569152E+20} & \dfrac{4.260420864E+22}{1.16187669216E+23} \\ -\dfrac{1.27486192896E+26}{1.394252030592E+26} & -\dfrac{5.416342272E+22}{9.29501353728E+22} & \dfrac{2.94425856E+21}{4.64750676864E+22} & \dfrac{1.550018304E+22}{1.54916892288E+23} & \dfrac{4.386016512E+22}{1.54916892288E+23} & \dfrac{3.028730112E+22}{9.29501353728E+22} \\ -\dfrac{5.688850176E+22}{1.54916892288E+23} & -\dfrac{7.5010597632E+25}{1.859002707456E+26} & -\dfrac{4.7884032E+20}{1.16187669216E+22} & \dfrac{3.732505344E+22}{6.19667569152E+23} & \dfrac{1.057536E+19}{3.09833784576E+22} & \dfrac{5.5570176E+20}{1.16187669216E+22} \\ \dfrac{6.301225728E+22}{4.841152884E+22} & \dfrac{9.545819904E+22}{9.29501353728E+22} & -\dfrac{8.070204672E+22}{4.64750676864E+23} & \dfrac{1.4457648384E+23}{7.7458446144E+23} & -\dfrac{1.9153269504E+23}{3.09833784576E+23} & -\dfrac{4.045250304E+22}{5.8093834608E+22} \\ -\dfrac{2.2878713088E+23}{1.54916892288E+24} & -\dfrac{1.8128155392E+23}{9.29501353728E+23} & \dfrac{1.6549632E+20}{2.32375338432E+21} & \dfrac{1.488538368E+22}{1.54916892288E+24} & \dfrac{1.289689344E+22}{1.239335138304E+23} & \dfrac{2.685973248E+22}{2.32375338432E+23} \\ \dfrac{3.662648064E+22}{6.19667569152E+23} & \dfrac{1.28173824E+21}{1.239335138304E+23} & -\dfrac{2.01164544E+21}{3.09833784576E+23} & -\dfrac{9.353219328E+22}{6.19667569152E+23} & \dfrac{1.51905024E+21}{1.239335138304E+22} & -\dfrac{1.12317696E+21}{3.09833784576E+22}\end{pmatrix}\) .
Précisément on a calculé \( A^{-1}_{5, 1}=B_{5, 1}=
\left(\dfrac{7.7458446144E+19}{23040000000000000}\right)^{-1}Co(A)_{1, 5}=
\left(\dfrac{7.7458446144E+19}{23040000000000000}\right)^{-1}\times(-1)^{1+5}\det\begin{pmatrix}-\dfrac{1}{4} & 2 & -\dfrac{14}{5} & 0 & -1 \\ -3 & 4 & 3 & \dfrac{11}{4} & -\dfrac{17}{5} \\ \dfrac{5}{4} & 2 & -3 & 2 & 1 \\ \dfrac{4}{3} & \dfrac{12}{5} & -4 & \dfrac{5}{3} & \dfrac{22}{3} \\ \dfrac{7}{5} & -\dfrac{21}{4} & -4 & -4 & -5\end{pmatrix}=-\dfrac{2.2878713088E+23}{1.54916892288E+24}\) .
- On observe que \( B^{-1}=\left(A^{-1}\right)^{-1}=A\) . Trivialement, et sans calcul,
\[ B^{-1}=\begin{pmatrix}1 & -3 & -1 & -\dfrac{1}{2} & 3 & 1 \\ -\dfrac{1}{4} & 2 & -\dfrac{14}{5} & 0 & -4 & -1 \\ -3 & 4 & 3 & \dfrac{11}{4} & \dfrac{25}{2} & -\dfrac{17}{5} \\ \dfrac{5}{4} & 2 & -3 & 2 & 4 & 1 \\ \dfrac{4}{3} & \dfrac{12}{5} & -4 & \dfrac{5}{3} & 3 & \dfrac{22}{3} \\ \dfrac{7}{5} & -\dfrac{21}{4} & -4 & -4 & -5 & -5\end{pmatrix}\]
- Le système
\( \left\{\begin{array}{*{7}{cr}}
&x&-&3y &-&z &-&\dfrac{1}{2}t &+&3u &+&v &=&-6\\
&-\dfrac{1}{4}x&+&2y &-&\dfrac{14}{5}z &&&-&4u &-&v &=&2\\
&-3x&+&4y &+&3z &+&\dfrac{11}{4}t &+&\dfrac{25}{2}u &-&\dfrac{17}{5}v &=&0\\
&\dfrac{5}{4}x&+&2y &-&3z &+&2t &+&4u &+&v &=&6\\
&\dfrac{4}{3}x&+&\dfrac{12}{5}y &-&4z &+&\dfrac{5}{3}t &+&3u &+&\dfrac{22}{3}v &=&-5\\
&\dfrac{7}{5}x&-&\dfrac{21}{4}y &-&4z &-&4t &-&5u &-&5v &=&9\\
\end{array}
\right.\) est équivalent à l'équation \( AX=a\) où la matrice \( A\) est celle de l'énoncé, \( X\) la matrice colonne des indéterminés et \( a=\begin{pmatrix}-6 \\ 2 \\ 0 \\ 6 \\ -5 \\ 9\end{pmatrix}\) de sorte que la solution est \( X=A^{-1}a=\begin{pmatrix}-\dfrac{1.6662645504E+23}{1.54916892288E+23} & -\dfrac{4.942165248E+22}{4.64750676864E+22} & -\dfrac{6.74572032E+21}{4.64750676864E+22} & \dfrac{8.920689408E+22}{1.54916892288E+23} & \dfrac{6.559488E+19}{6.19667569152E+20} & \dfrac{4.260420864E+22}{1.16187669216E+23} \\ -\dfrac{1.27486192896E+26}{1.394252030592E+26} & -\dfrac{5.416342272E+22}{9.29501353728E+22} & \dfrac{2.94425856E+21}{4.64750676864E+22} & \dfrac{1.550018304E+22}{1.54916892288E+23} & \dfrac{4.386016512E+22}{1.54916892288E+23} & \dfrac{3.028730112E+22}{9.29501353728E+22} \\ -\dfrac{5.688850176E+22}{1.54916892288E+23} & -\dfrac{7.5010597632E+25}{1.859002707456E+26} & -\dfrac{4.7884032E+20}{1.16187669216E+22} & \dfrac{3.732505344E+22}{6.19667569152E+23} & \dfrac{1.057536E+19}{3.09833784576E+22} & \dfrac{5.5570176E+20}{1.16187669216E+22} \\ \dfrac{6.301225728E+22}{4.841152884E+22} & \dfrac{9.545819904E+22}{9.29501353728E+22} & -\dfrac{8.070204672E+22}{4.64750676864E+23} & \dfrac{1.4457648384E+23}{7.7458446144E+23} & -\dfrac{1.9153269504E+23}{3.09833784576E+23} & -\dfrac{4.045250304E+22}{5.8093834608E+22} \\ -\dfrac{2.2878713088E+23}{1.54916892288E+24} & -\dfrac{1.8128155392E+23}{9.29501353728E+23} & \dfrac{1.6549632E+20}{2.32375338432E+21} & \dfrac{1.488538368E+22}{1.54916892288E+24} & \dfrac{1.289689344E+22}{1.239335138304E+23} & \dfrac{2.685973248E+22}{2.32375338432E+23} \\ \dfrac{3.662648064E+22}{6.19667569152E+23} & \dfrac{1.28173824E+21}{1.239335138304E+23} & -\dfrac{2.01164544E+21}{3.09833784576E+23} & -\dfrac{9.353219328E+22}{6.19667569152E+23} & \dfrac{1.51905024E+21}{1.239335138304E+22} & -\dfrac{1.12317696E+21}{3.09833784576E+22}\end{pmatrix}\times\begin{pmatrix}-6 \\ 2 \\ 0 \\ 6 \\ -5 \\ 9\end{pmatrix}=\begin{pmatrix}\dfrac{8.4741617308791E+113}{8.0303854342143E+112} \\ \dfrac{1.8612200494694E+119}{2.8909387563171E+118} \\ \dfrac{1.4046541619321E+118}{6.4243083473714E+117} \\ -\dfrac{4.9009115161649E+116}{6.2737386204799E+115} \\ \dfrac{6.8975670403385E+118}{6.4243083473714E+118} \\ -\dfrac{3.9812647152143E+115}{1.8273588188079E+115}\end{pmatrix}\) . Ainsi \( x=\dfrac{8.4741617308791E+113}{8.0303854342143E+112}\) , \( y=\dfrac{1.8612200494694E+119}{2.8909387563171E+118}\) , \( z=\dfrac{1.4046541619321E+118}{6.4243083473714E+117}\) , \( t=\dfrac{4.9009115161649E+116}{6.2737386204799E+115}\) , \( u=\dfrac{6.8975670403385E+118}{6.4243083473714E+118}\) et \( v=\dfrac{3.9812647152143E+115}{1.8273588188079E+115}\)
- Le système
\( \left\{\begin{array}{*{7}{cr}}
&-\dfrac{1.6662645504E+23}{1.54916892288E+23}x&-&\dfrac{4.942165248E+22}{4.64750676864E+22}y &-&\dfrac{6.74572032E+21}{4.64750676864E+22}z &+&\dfrac{8.920689408E+22}{1.54916892288E+23}t &+&\dfrac{6.559488E+19}{6.19667569152E+20}u &+&\dfrac{4.260420864E+22}{1.16187669216E+23}v &=&7\\
&-\dfrac{1.27486192896E+26}{1.394252030592E+26}x&-&\dfrac{5.416342272E+22}{9.29501353728E+22}y &+&\dfrac{2.94425856E+21}{4.64750676864E+22}z &+&\dfrac{1.550018304E+22}{1.54916892288E+23}t &+&\dfrac{4.386016512E+22}{1.54916892288E+23}u &+&\dfrac{3.028730112E+22}{9.29501353728E+22}v &=&-4\\
&-\dfrac{5.688850176E+22}{1.54916892288E+23}x&-&\dfrac{7.5010597632E+25}{1.859002707456E+26}y &-&\dfrac{4.7884032E+20}{1.16187669216E+22}z &+&\dfrac{3.732505344E+22}{6.19667569152E+23}t &+&\dfrac{1.057536E+19}{3.09833784576E+22}u &+&\dfrac{5.5570176E+20}{1.16187669216E+22}v &=&6\\
&\dfrac{6.301225728E+22}{4.841152884E+22}x&+&\dfrac{9.545819904E+22}{9.29501353728E+22}y &-&\dfrac{8.070204672E+22}{4.64750676864E+23}z &+&\dfrac{1.4457648384E+23}{7.7458446144E+23}t &-&\dfrac{1.9153269504E+23}{3.09833784576E+23}u &-&\dfrac{4.045250304E+22}{5.8093834608E+22}v &=&9\\
&-\dfrac{2.2878713088E+23}{1.54916892288E+24}x&-&\dfrac{1.8128155392E+23}{9.29501353728E+23}y &+&\dfrac{1.6549632E+20}{2.32375338432E+21}z &+&\dfrac{1.488538368E+22}{1.54916892288E+24}t &+&\dfrac{1.289689344E+22}{1.239335138304E+23}u &+&\dfrac{2.685973248E+22}{2.32375338432E+23}v &=&-\dfrac{36}{7}\\
&\dfrac{3.662648064E+22}{6.19667569152E+23}x&+&\dfrac{1.28173824E+21}{1.239335138304E+23}y &-&\dfrac{2.01164544E+21}{3.09833784576E+23}z &-&\dfrac{9.353219328E+22}{6.19667569152E+23}t &+&\dfrac{1.51905024E+21}{1.239335138304E+22}u &-&\dfrac{1.12317696E+21}{3.09833784576E+22}v &=&-9\\
\end{array}
\right.\) est équivalent à l'équation \( BX=b\) où la matrice \( B=A^{-1}\) déterminée précédement, \( X\) la matrice colonne des indéterminés et \( b=\begin{pmatrix}7 \\ -4 \\ 6 \\ 9 \\ -\dfrac{36}{7} \\ -9\end{pmatrix}\) de sorte que la solution est \( X=B^{-1}b=Ab=\begin{pmatrix}1 & -3 & -1 & -\dfrac{1}{2} & 3 & 1 \\ -\dfrac{1}{4} & 2 & -\dfrac{14}{5} & 0 & -4 & -1 \\ -3 & 4 & 3 & \dfrac{11}{4} & \dfrac{25}{2} & -\dfrac{17}{5} \\ \dfrac{5}{4} & 2 & -3 & 2 & 4 & 1 \\ \dfrac{4}{3} & \dfrac{12}{5} & -4 & \dfrac{5}{3} & 3 & \dfrac{22}{3} \\ \dfrac{7}{5} & -\dfrac{21}{4} & -4 & -4 & -5 & -5\end{pmatrix}\times\begin{pmatrix}7 \\ -4 \\ 6 \\ 9 \\ -\dfrac{36}{7} \\ -9\end{pmatrix}=\begin{pmatrix}-\dfrac{223}{14} \\ \dfrac{423}{140} \\ -\dfrac{3911}{140} \\ -\dfrac{807}{28} \\ -\dfrac{9523}{105} \\ \dfrac{1453}{35}\end{pmatrix}\) . Ainsi \( x=\dfrac{223}{14}\) , \( y=\dfrac{423}{140}\) , \( z=\dfrac{3911}{140}\) , \( t=\dfrac{807}{28}\) , \( u=\dfrac{9523}{105}\) et \( v=\dfrac{1453}{35}\)
