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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.

Exercice


On considère la matrice \[ A= \begin{pmatrix}1 & 5 & 2 & 4 & -4 & -4 \\ 4 & \dfrac{7}{4} & \dfrac{23}{3} & -2 & 3 & -2 \\ 1 & 0 & -5 & -2 & -\dfrac{5}{3} & -\dfrac{9}{4} \\ -5 & -4 & \dfrac{24}{5} & \dfrac{16}{3} & 2 & -4 \\ \dfrac{9}{2} & 0 & 4 & -5 & 0 & \dfrac{16}{3} \\ \dfrac{5}{4} & -1 & -\dfrac{11}{2} & 4 & 2 & -2\end{pmatrix}\]
  1. Donner les mineurs d'ordre \( (6, 1)\) et \( (3, 6)\) : \( \widehat{A}_{6, 1}=\) \( \widehat{A}_{3, 6}=\)
  2. Expliquer pourquoi \( \det(A)=\det \begin{pmatrix}1 & 5 & 2 & 4 & -4 & -4 \\ 0 & -\dfrac{73}{4} & -\dfrac{1}{3} & -18 & 19 & 14 \\ 0 & -5 & -7 & -6 & \dfrac{7}{3} & \dfrac{7}{4} \\ 0 & 21 & \dfrac{74}{5} & \dfrac{76}{3} & -18 & -24 \\ 0 & -\dfrac{45}{2} & -5 & -23 & 18 & \dfrac{70}{3} \\ 0 & -\dfrac{29}{4} & -8 & -1 & 7 & 3\end{pmatrix} \)
  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 \( A^{-1}_{5, 1}\) .
  6. Donner \( B^{-1}\) l'inverse de la matrice \( B\) . Justifier.
  7. Résoudre le système suivant. \( \left\{\begin{array}{*{7}{cr}} &x&+&5y &+&2z &+&4t &-&4u &-&4v &=&6\\ &4x&+&\dfrac{7}{4}y &+&\dfrac{23}{3}z &-&2t &+&3u &-&2v &=&8\\ &x&&&-&5z &-&2t &-&\dfrac{5}{3}u &-&\dfrac{9}{4}v &=&0\\ &-5x&-&4y &+&\dfrac{24}{5}z &+&\dfrac{16}{3}t &+&2u &-&4v &=&-2\\ &\dfrac{9}{2}x&&&+&4z &-&5t &&&+&\dfrac{16}{3}v &=&-9\\ &\dfrac{5}{4}x&-&y &-&\dfrac{11}{2}z &+&4t &+&2u &-&2v &=&-2\\ \end{array} \right. \)
  8. Résoudre le système suivant. \( \left\{\begin{array}{*{7}{cr}} &\dfrac{4.2283122751079E+24}{5.809275203578E+25}x&+&\dfrac{1.9067121487215E+20}{1.7929861739438E+22}y &+&\dfrac{4.2357477565774E+20}{4.4824654348595E+21}z &+&\dfrac{5.2047824798477E+19}{8.9649308697191E+20}t &+&\dfrac{1.0669232147734E+22}{5.3789585218315E+22}u &+&\dfrac{2.4255197155832E+22}{1.6136875565494E+23}v &=&-8\\ &-\dfrac{2.043265990199E+19}{8.0684377827472E+21}x&+&\dfrac{1.0151846651685E+21}{1.3447396304579E+22}y &-&\dfrac{2.5009403808651E+21}{1.3447396304579E+22}z &-&\dfrac{1.0888123277649E+21}{5.3789585218315E+21}t &-&\dfrac{2.065878949301E+20}{8.9649308697191E+20}u &-&\dfrac{2.8589160862041E+21}{4.0342188913736E+22}v &=&-1\\ &\dfrac{2.9758026040416E+20}{9.5625929277004E+21}x&+&\dfrac{4.4129817825212E+20}{2.1515834087326E+22}y &-&\dfrac{2.762318417783E+19}{1.7929861739438E+21}z &+&\dfrac{5.1654412205914E+19}{8.9649308697191E+20}t &+&\dfrac{3.8101678990929E+20}{7.1719446957753E+21}u &-&\dfrac{8.3877383027969E+20}{2.1515834087326E+22}v &=&-4\\ &\dfrac{7.3305944589058E+24}{9.2948403257247E+25}x&-&\dfrac{2.1830547411036E+21}{3.5859723478876E+22}y &-&\dfrac{3.0338888162377E+20}{2.9883102899064E+21}z &+&\dfrac{2.2117226175441E+20}{5.3789585218315E+21}t &+&\dfrac{2.8120900697245E+20}{3.9844137198751E+21}u &+&\dfrac{1.4741781451155E+21}{1.1953241159625E+22}v &=&-5\\ &-\dfrac{4.0193407726232E+24}{3.0982801085749E+25}x&+&\dfrac{4.5839724498007E+21}{3.5859723478876E+22}y &-&\dfrac{4.1612324361566E+21}{2.6894792609157E+22}z &-&\dfrac{8.2429360208283E+20}{7.1719446957753E+21}t &-&\dfrac{1.3467818908767E+21}{7.1719446957753E+21}u &+&\dfrac{3.7432497394931E+21}{1.0757917043663E+23}v &=&-9\\ &-\dfrac{1.9389455334063E+20}{1.7929861739438E+22}x&-&\dfrac{3.6499927721207E+20}{4.4824654348595E+21}y &-&\dfrac{2.4407945380374E+20}{1.4941551449532E+21}z &-&\dfrac{9.6195990858048E+19}{1.7929861739438E+21}t &+&\dfrac{6.2480366375973E+20}{1.3447396304579E+22}u &+&\dfrac{8.0842981783565E+19}{4.4824654348595E+21}v &=&2\\ \end{array} \right. \)
Cliquer ici pour afficher la solution

Exercice


  1. \( \widehat{A}_{6, 1}=\begin{pmatrix}5 & 2 & 4 & -4 & -4 \\ \dfrac{7}{4} & \dfrac{23}{3} & -2 & 3 & -2 \\ 0 & -5 & -2 & -\dfrac{5}{3} & -\dfrac{9}{4} \\ -4 & \dfrac{24}{5} & \dfrac{16}{3} & 2 & -4 \\ 0 & 4 & -5 & 0 & \dfrac{16}{3}\end{pmatrix}\) \( \widehat{A}_{3, 6}=\begin{pmatrix}1 & 5 & 2 & 4 & -4 \\ 4 & \dfrac{7}{4} & \dfrac{23}{3} & -2 & 3 \\ -5 & -4 & \dfrac{24}{5} & \dfrac{16}{3} & 2 \\ \dfrac{9}{2} & 0 & 4 & -5 & 0 \\ \dfrac{5}{4} & -1 & -\dfrac{11}{2} & 4 & 2\end{pmatrix}\)
  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 : \( L_{2}\leftarrow L_{2}-\left(4\right)L_1\) , \( L_{3}\leftarrow L_{3}-\left(1\right)L_1\) , \( L_{4}\leftarrow L_{4}-\left(-5\right)L_1\) , \( L_{5}\leftarrow L_{5}-\left(\dfrac{9}{2}\right)L_1\) et \( L_{6}\leftarrow L_{6}-\left(\dfrac{5}{4}\right)L_1\)
  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 & 5 & 2 & 4 & -4 & -4 \\ 0 & -\dfrac{73}{4} & -\dfrac{1}{3} & -18 & 19 & 14 \\ 0 & -5 & -7 & -6 & \dfrac{7}{3} & \dfrac{7}{4} \\ 0 & 21 & \dfrac{74}{5} & \dfrac{76}{3} & -18 & -24 \\ 0 & -\dfrac{45}{2} & -5 & -23 & 18 & \dfrac{70}{3} \\ 0 & -\dfrac{29}{4} & -8 & -1 & 7 & 3\end{pmatrix}\\ &=&1\times\det\begin{pmatrix}-\dfrac{73}{4} & -\dfrac{1}{3} & -18 & 19 & 14 \\ -5 & -7 & -6 & \dfrac{7}{3} & \dfrac{7}{4} \\ 21 & \dfrac{74}{5} & \dfrac{76}{3} & -18 & -24 \\ -\dfrac{45}{2} & -5 & -23 & 18 & \dfrac{70}{3} \\ -\dfrac{29}{4} & -8 & -1 & 7 & 3\end{pmatrix}\\ &=&-\dfrac{9.9610342996879E+19}{1652994086400000} \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{9.9610342996879E+19}{1652994086400000}\right)^{-1}{}^tCo\begin{pmatrix}1 & 5 & 2 & 4 & -4 & -4 \\ 4 & \dfrac{7}{4} & \dfrac{23}{3} & -2 & 3 & -2 \\ 1 & 0 & -5 & -2 & -\dfrac{5}{3} & -\dfrac{9}{4} \\ -5 & -4 & \dfrac{24}{5} & \dfrac{16}{3} & 2 & -4 \\ \dfrac{9}{2} & 0 & 4 & -5 & 0 & \dfrac{16}{3} \\ \dfrac{5}{4} & -1 & -\dfrac{11}{2} & 4 & 2 & -2\end{pmatrix} =-\dfrac{1652994086400000}{9.9610342996879E+19}\begin{pmatrix}-\dfrac{4.2118363602141E+44}{9.6026975577845E+40} & -\dfrac{1.8992825113047E+40}{2.9637955425261E+37} & -\dfrac{4.2192428688094E+40}{7.4094888563152E+36} & -\dfrac{5.1845016804177E+39}{1.481897771263E+36} & -\dfrac{1.0627658737491E+42}{8.8913866275783E+37} & -\dfrac{2.4160685081493E+42}{2.6674159882735E+38} \\ \dfrac{2.0353042611758E+39}{1.3337079941367E+37} & -\dfrac{1.011228927026E+41}{2.2228466568946E+37} & \dfrac{2.4911952915272E+41}{2.2228466568946E+37} & \dfrac{1.0845696942789E+41}{8.8913866275783E+36} & \dfrac{2.057829107299E+40}{1.481897771263E+36} & \dfrac{2.8477761194608E+41}{6.6685399706837E+37} \\ -\dfrac{2.9642071807959E+40}{1.5806909560139E+37} & -\dfrac{4.3957862899592E+40}{3.5565546510313E+37} & \dfrac{2.7515548506196E+39}{2.9637955425261E+36} & -\dfrac{5.1453137171332E+39}{1.481897771263E+36} & -\dfrac{3.7953213130434E+40}{1.1855182170104E+37} & \dfrac{8.3550548930966E+40}{3.5565546510313E+37} \\ -\dfrac{7.3020302842262E+44}{1.5364316092455E+41} & \dfrac{2.1745483154229E+41}{5.9275910850522E+37} & \dfrac{3.0220670559983E+40}{4.9396592375435E+36} & -\dfrac{2.2031044854752E+40}{8.8913866275783E+36} & -\dfrac{2.8011325638338E+40}{6.5862123167246E+36} & -\dfrac{1.4684339067346E+41}{1.9758636950174E+37} \\ \dfrac{4.0036791298233E+44}{5.1214386974851E+40} & -\dfrac{4.5661106801289E+41}{5.9275910850522E+37} & \dfrac{4.1450179025529E+41}{4.4456933137891E+37} & \dfrac{8.2108168433603E+40}{1.1855182170104E+37} & \dfrac{1.3415340609221E+41}{1.1855182170104E+37} & -\dfrac{3.7286639047389E+41}{1.7782773255157E+38} \\ \dfrac{1.9313902963487E+40}{2.9637955425261E+37} & \dfrac{3.6357703196707E+40}{7.4094888563152E+36} & \dfrac{2.4312838111881E+40}{2.4698296187717E+36} & \dfrac{9.5821156442948E+39}{2.9637955425261E+36} & -\dfrac{6.2236907252813E+40}{2.2228466568946E+37} & -\dfrac{8.0527971443513E+39}{7.4094888563152E+36}\end{pmatrix} =\begin{pmatrix}\dfrac{4.2283122751079E+24}{5.809275203578E+25} & \dfrac{1.9067121487215E+20}{1.7929861739438E+22} & \dfrac{4.2357477565774E+20}{4.4824654348595E+21} & \dfrac{5.2047824798477E+19}{8.9649308697191E+20} & \dfrac{1.0669232147734E+22}{5.3789585218315E+22} & \dfrac{2.4255197155832E+22}{1.6136875565494E+23} \\ -\dfrac{2.043265990199E+19}{8.0684377827472E+21} & \dfrac{1.0151846651685E+21}{1.3447396304579E+22} & -\dfrac{2.5009403808651E+21}{1.3447396304579E+22} & -\dfrac{1.0888123277649E+21}{5.3789585218315E+21} & -\dfrac{2.065878949301E+20}{8.9649308697191E+20} & -\dfrac{2.8589160862041E+21}{4.0342188913736E+22} \\ \dfrac{2.9758026040416E+20}{9.5625929277004E+21} & \dfrac{4.4129817825212E+20}{2.1515834087326E+22} & -\dfrac{2.762318417783E+19}{1.7929861739438E+21} & \dfrac{5.1654412205914E+19}{8.9649308697191E+20} & \dfrac{3.8101678990929E+20}{7.1719446957753E+21} & -\dfrac{8.3877383027969E+20}{2.1515834087326E+22} \\ \dfrac{7.3305944589058E+24}{9.2948403257247E+25} & -\dfrac{2.1830547411036E+21}{3.5859723478876E+22} & -\dfrac{3.0338888162377E+20}{2.9883102899064E+21} & \dfrac{2.2117226175441E+20}{5.3789585218315E+21} & \dfrac{2.8120900697245E+20}{3.9844137198751E+21} & \dfrac{1.4741781451155E+21}{1.1953241159625E+22} \\ -\dfrac{4.0193407726232E+24}{3.0982801085749E+25} & \dfrac{4.5839724498007E+21}{3.5859723478876E+22} & -\dfrac{4.1612324361566E+21}{2.6894792609157E+22} & -\dfrac{8.2429360208283E+20}{7.1719446957753E+21} & -\dfrac{1.3467818908767E+21}{7.1719446957753E+21} & \dfrac{3.7432497394931E+21}{1.0757917043663E+23} \\ -\dfrac{1.9389455334063E+20}{1.7929861739438E+22} & -\dfrac{3.6499927721207E+20}{4.4824654348595E+21} & -\dfrac{2.4407945380374E+20}{1.4941551449532E+21} & -\dfrac{9.6195990858048E+19}{1.7929861739438E+21} & \dfrac{6.2480366375973E+20}{1.3447396304579E+22} & \dfrac{8.0842981783565E+19}{4.4824654348595E+21}\end{pmatrix}\) . Précisément on a calculé \( A^{-1}_{5, 1}=B_{5, 1}= \left(-\dfrac{9.9610342996879E+19}{1652994086400000}\right)^{-1}Co(A)_{1, 5}= \left(-\dfrac{9.9610342996879E+19}{1652994086400000}\right)^{-1}\times(-1)^{1+5}\det\begin{pmatrix}4 & \dfrac{7}{4} & \dfrac{23}{3} & -2 & -2 \\ 1 & 0 & -5 & -2 & -\dfrac{9}{4} \\ -5 & -4 & \dfrac{24}{5} & \dfrac{16}{3} & -4 \\ \dfrac{9}{2} & 0 & 4 & -5 & \dfrac{16}{3} \\ \dfrac{5}{4} & -1 & -\dfrac{11}{2} & 4 & -2\end{pmatrix}=-\dfrac{4.0193407726232E+24}{3.0982801085749E+25}\) .
  6. On observe que \( B^{-1}=\left(A^{-1}\right)^{-1}=A\) . Trivialement, et sans calcul, \[ B^{-1}=\begin{pmatrix}1 & 5 & 2 & 4 & -4 & -4 \\ 4 & \dfrac{7}{4} & \dfrac{23}{3} & -2 & 3 & -2 \\ 1 & 0 & -5 & -2 & -\dfrac{5}{3} & -\dfrac{9}{4} \\ -5 & -4 & \dfrac{24}{5} & \dfrac{16}{3} & 2 & -4 \\ \dfrac{9}{2} & 0 & 4 & -5 & 0 & \dfrac{16}{3} \\ \dfrac{5}{4} & -1 & -\dfrac{11}{2} & 4 & 2 & -2\end{pmatrix}\]
  7. Le système \( \left\{\begin{array}{*{7}{cr}} &x&+&5y &+&2z &+&4t &-&4u &-&4v &=&6\\ &4x&+&\dfrac{7}{4}y &+&\dfrac{23}{3}z &-&2t &+&3u &-&2v &=&8\\ &x&&&-&5z &-&2t &-&\dfrac{5}{3}u &-&\dfrac{9}{4}v &=&0\\ &-5x&-&4y &+&\dfrac{24}{5}z &+&\dfrac{16}{3}t &+&2u &-&4v &=&-2\\ &\dfrac{9}{2}x&&&+&4z &-&5t &&&+&\dfrac{16}{3}v &=&-9\\ &\dfrac{5}{4}x&-&y &-&\dfrac{11}{2}z &+&4t &+&2u &-&2v &=&-2\\ \end{array} \right.\) est équivalent à l'équation \( AX=a\) où la matrice \( A\) est celle de l'énoncé, \( X\) la matrice colonne des indéterminés et \( a=\begin{pmatrix}6 \\ 8 \\ 0 \\ -2 \\ -9 \\ -2\end{pmatrix}\) de sorte que la solution est \( X=A^{-1}a=\begin{pmatrix}\dfrac{4.2283122751079E+24}{5.809275203578E+25} & \dfrac{1.9067121487215E+20}{1.7929861739438E+22} & \dfrac{4.2357477565774E+20}{4.4824654348595E+21} & \dfrac{5.2047824798477E+19}{8.9649308697191E+20} & \dfrac{1.0669232147734E+22}{5.3789585218315E+22} & \dfrac{2.4255197155832E+22}{1.6136875565494E+23} \\ -\dfrac{2.043265990199E+19}{8.0684377827472E+21} & \dfrac{1.0151846651685E+21}{1.3447396304579E+22} & -\dfrac{2.5009403808651E+21}{1.3447396304579E+22} & -\dfrac{1.0888123277649E+21}{5.3789585218315E+21} & -\dfrac{2.065878949301E+20}{8.9649308697191E+20} & -\dfrac{2.8589160862041E+21}{4.0342188913736E+22} \\ \dfrac{2.9758026040416E+20}{9.5625929277004E+21} & \dfrac{4.4129817825212E+20}{2.1515834087326E+22} & -\dfrac{2.762318417783E+19}{1.7929861739438E+21} & \dfrac{5.1654412205914E+19}{8.9649308697191E+20} & \dfrac{3.8101678990929E+20}{7.1719446957753E+21} & -\dfrac{8.3877383027969E+20}{2.1515834087326E+22} \\ \dfrac{7.3305944589058E+24}{9.2948403257247E+25} & -\dfrac{2.1830547411036E+21}{3.5859723478876E+22} & -\dfrac{3.0338888162377E+20}{2.9883102899064E+21} & \dfrac{2.2117226175441E+20}{5.3789585218315E+21} & \dfrac{2.8120900697245E+20}{3.9844137198751E+21} & \dfrac{1.4741781451155E+21}{1.1953241159625E+22} \\ -\dfrac{4.0193407726232E+24}{3.0982801085749E+25} & \dfrac{4.5839724498007E+21}{3.5859723478876E+22} & -\dfrac{4.1612324361566E+21}{2.6894792609157E+22} & -\dfrac{8.2429360208283E+20}{7.1719446957753E+21} & -\dfrac{1.3467818908767E+21}{7.1719446957753E+21} & \dfrac{3.7432497394931E+21}{1.0757917043663E+23} \\ -\dfrac{1.9389455334063E+20}{1.7929861739438E+22} & -\dfrac{3.6499927721207E+20}{4.4824654348595E+21} & -\dfrac{2.4407945380374E+20}{1.4941551449532E+21} & -\dfrac{9.6195990858048E+19}{1.7929861739438E+21} & \dfrac{6.2480366375973E+20}{1.3447396304579E+22} & \dfrac{8.0842981783565E+19}{4.4824654348595E+21}\end{pmatrix}\times\begin{pmatrix}6 \\ 8 \\ 0 \\ -2 \\ -9 \\ -2\end{pmatrix}=\begin{pmatrix}-\dfrac{1.3617590503499E+115}{8.1051952630596E+114} \\ \dfrac{6.7739267075539E+109}{2.1107279330884E+109} \\ -\dfrac{4.6850805134369E+108}{2.8462688440922E+109} \\ -\dfrac{8.3501483432417E+113}{8.5388065322767E+113} \\ \dfrac{1.2877568385598E+115}{6.1479407032392E+114} \\ -\dfrac{9.2354655837842E+108}{8.6861231814339E+108}\end{pmatrix}\) . Ainsi \( x=\dfrac{1.3617590503499E+115}{8.1051952630596E+114}\) , \( y=\dfrac{6.7739267075539E+109}{2.1107279330884E+109}\) , \( z=\dfrac{4.6850805134369E+108}{2.8462688440922E+109}\) , \( t=\dfrac{8.3501483432417E+113}{8.5388065322767E+113}\) , \( u=\dfrac{1.2877568385598E+115}{6.1479407032392E+114}\) et \( v=\dfrac{9.2354655837842E+108}{8.6861231814339E+108}\)
  8. Le système \( \left\{\begin{array}{*{7}{cr}} &\dfrac{4.2283122751079E+24}{5.809275203578E+25}x&+&\dfrac{1.9067121487215E+20}{1.7929861739438E+22}y &+&\dfrac{4.2357477565774E+20}{4.4824654348595E+21}z &+&\dfrac{5.2047824798477E+19}{8.9649308697191E+20}t &+&\dfrac{1.0669232147734E+22}{5.3789585218315E+22}u &+&\dfrac{2.4255197155832E+22}{1.6136875565494E+23}v &=&-8\\ &-\dfrac{2.043265990199E+19}{8.0684377827472E+21}x&+&\dfrac{1.0151846651685E+21}{1.3447396304579E+22}y &-&\dfrac{2.5009403808651E+21}{1.3447396304579E+22}z &-&\dfrac{1.0888123277649E+21}{5.3789585218315E+21}t &-&\dfrac{2.065878949301E+20}{8.9649308697191E+20}u &-&\dfrac{2.8589160862041E+21}{4.0342188913736E+22}v &=&-1\\ &\dfrac{2.9758026040416E+20}{9.5625929277004E+21}x&+&\dfrac{4.4129817825212E+20}{2.1515834087326E+22}y &-&\dfrac{2.762318417783E+19}{1.7929861739438E+21}z &+&\dfrac{5.1654412205914E+19}{8.9649308697191E+20}t &+&\dfrac{3.8101678990929E+20}{7.1719446957753E+21}u &-&\dfrac{8.3877383027969E+20}{2.1515834087326E+22}v &=&-4\\ &\dfrac{7.3305944589058E+24}{9.2948403257247E+25}x&-&\dfrac{2.1830547411036E+21}{3.5859723478876E+22}y &-&\dfrac{3.0338888162377E+20}{2.9883102899064E+21}z &+&\dfrac{2.2117226175441E+20}{5.3789585218315E+21}t &+&\dfrac{2.8120900697245E+20}{3.9844137198751E+21}u &+&\dfrac{1.4741781451155E+21}{1.1953241159625E+22}v &=&-5\\ &-\dfrac{4.0193407726232E+24}{3.0982801085749E+25}x&+&\dfrac{4.5839724498007E+21}{3.5859723478876E+22}y &-&\dfrac{4.1612324361566E+21}{2.6894792609157E+22}z &-&\dfrac{8.2429360208283E+20}{7.1719446957753E+21}t &-&\dfrac{1.3467818908767E+21}{7.1719446957753E+21}u &+&\dfrac{3.7432497394931E+21}{1.0757917043663E+23}v &=&-9\\ &-\dfrac{1.9389455334063E+20}{1.7929861739438E+22}x&-&\dfrac{3.6499927721207E+20}{4.4824654348595E+21}y &-&\dfrac{2.4407945380374E+20}{1.4941551449532E+21}z &-&\dfrac{9.6195990858048E+19}{1.7929861739438E+21}t &+&\dfrac{6.2480366375973E+20}{1.3447396304579E+22}u &+&\dfrac{8.0842981783565E+19}{4.4824654348595E+21}v &=&2\\ \end{array} \right.\) est équivalent à l'équation \( BX=b\) où la matrice \( B=A^{-1}\) déterminée précédement, \( X\) la matrice colonne des indéterminés et \( b=\begin{pmatrix}-8 \\ -1 \\ -4 \\ -5 \\ -9 \\ 2\end{pmatrix}\) de sorte que la solution est \( X=B^{-1}b=Ab=\begin{pmatrix}1 & 5 & 2 & 4 & -4 & -4 \\ 4 & \dfrac{7}{4} & \dfrac{23}{3} & -2 & 3 & -2 \\ 1 & 0 & -5 & -2 & -\dfrac{5}{3} & -\dfrac{9}{4} \\ -5 & -4 & \dfrac{24}{5} & \dfrac{16}{3} & 2 & -4 \\ \dfrac{9}{2} & 0 & 4 & -5 & 0 & \dfrac{16}{3} \\ \dfrac{5}{4} & -1 & -\dfrac{11}{2} & 4 & 2 & -2\end{pmatrix}\times\begin{pmatrix}-8 \\ -1 \\ -4 \\ -5 \\ -9 \\ 2\end{pmatrix}=\begin{pmatrix}-13 \\ -\dfrac{1025}{12} \\ \dfrac{65}{2} \\ -\dfrac{418}{15} \\ -\dfrac{49}{3} \\ -29\end{pmatrix}\) . Ainsi \( x=13\) , \( y=\dfrac{1025}{12}\) , \( z=\dfrac{65}{2}\) , \( t=\dfrac{418}{15}\) , \( u=\dfrac{49}{3}\) et \( v=29\)