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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 \[ A= \begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -5 & -4 \\ -1 & 2 & -5 & -5 & 0 & -5 \\ -3 & 4 & 4 & \dfrac{6}{5} & 3 & 2 \\ -4 & 3 & 2 & \dfrac{8}{5} & -4 & -\dfrac{17}{5} \\ -\dfrac{7}{4} & \dfrac{10}{3} & -\dfrac{23}{5} & \dfrac{23}{2} & -3 & 1 \\ \dfrac{16}{5} & -2 & 2 & 3 & 0 & \dfrac{8}{3}\end{pmatrix}\]
  1. Donner les mineurs d'ordre \( (4, 5)\) et \( (4, 6)\) : \( \widehat{A}_{4, 5}=\) \( \widehat{A}_{4, 6}=\)
  2. Expliquer pourquoi \( \det(A)=\det \begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -5 & -4 \\ 0 & 1 & -\dfrac{7}{4} & -\dfrac{31}{2} & -5 & -9 \\ 0 & 1 & \dfrac{55}{4} & -\dfrac{303}{10} & -12 & -10 \\ 0 & -1 & 15 & -\dfrac{202}{5} & -24 & -\dfrac{97}{5} \\ 0 & \dfrac{19}{12} & \dfrac{87}{80} & -\dfrac{55}{8} & -\dfrac{47}{4} & -6 \\ 0 & \dfrac{6}{5} & -\dfrac{42}{5} & \dfrac{183}{5} & 16 & \dfrac{232}{15}\end{pmatrix} \)
  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 \( A^{-1}_{6, 4}\) .
  6. Donner \( B^{-1}\) l'inverse de la matrice \( B\) . Justifier.
  7. Résoudre le système suivant. \( \left\{\begin{array}{*{7}{cr}} &x&-&y &+&\dfrac{13}{4}z &-&\dfrac{21}{2}t &-&5u &-&4v &=&-6\\ &-x&+&2y &-&5z &-&5t &&&-&5v &=&3\\ &-3x&+&4y &+&4z &+&\dfrac{6}{5}t &+&3u &+&2v &=&7\\ &-4x&+&3y &+&2z &+&\dfrac{8}{5}t &-&4u &-&\dfrac{17}{5}v &=&-4\\ &-\dfrac{7}{4}x&+&\dfrac{10}{3}y &-&\dfrac{23}{5}z &+&\dfrac{23}{2}t &-&3u &+&v &=&-7\\ &\dfrac{16}{5}x&-&2y &+&2z &+&3t &&&+&\dfrac{8}{3}v &=&-5\\ \end{array} \right. \)
  8. Résoudre le système suivant. \( \left\{\begin{array}{*{7}{cr}} &\dfrac{4878306000000}{92194714147500}x&+&\dfrac{122801589000000}{460973570737500}y &+&\dfrac{36910323000000}{368778856590000}z &-&\dfrac{5816367000000}{368778856590000}t &+&\dfrac{4047813000000}{122926285530000}u &+&\dfrac{579399399000000}{1229262855300000}v &=&-9\\ &\dfrac{28366767000000}{204877142550000}x&+&\dfrac{2324428569000000}{12292628553000000}y &+&\dfrac{29836917000000}{122926285530000}z &-&\dfrac{16439031000000}{163901714040000}t &+&\dfrac{17909181000000}{122926285530000}u &+&\dfrac{161997813000000}{819508570200000}v &=&0\\ &-\dfrac{107534871000000}{3073157138250000}x&+&\dfrac{13883157000000}{512192856375000}y &+&\dfrac{74542113000000}{1843894282950000}z &+&\dfrac{7364196000000}{51219285637500}t &-&\dfrac{14283702000000}{153657856912500}u &+&\dfrac{228877731000000}{1229262855300000}v &=&9\\ &-\dfrac{243410346000000}{2304867853687500}x&+&\dfrac{107554149000000}{2048771425500000}y &-&\dfrac{1236303000000}{30731571382500}z &+&\dfrac{490094631000000}{3687788565900000}t &-&\dfrac{12726801000000}{307315713825000}u &+&\dfrac{381479301000000}{2458525710600000}v &=&5\\ &-\dfrac{1278417087000000}{7682892845625000}x&+&\dfrac{1482397939125000000}{1.1524339268437E+19}y &+&\dfrac{764566533000000}{36877885659000000}z &+&\dfrac{757741959000000}{9219471414750000}t &-&\dfrac{148389327000000}{1024385712750000}u &+&\dfrac{830683269000000}{6146314276500000}v &=&-9\\ &\dfrac{569760561000000}{3073157138250000}x&-&\dfrac{3162216753000000}{12292628553000000}y &+&\dfrac{23624541000000}{307315713825000}z &-&\dfrac{48193137000000}{153657856912500}t &+&\dfrac{38119653000000}{204877142550000}u &-&\dfrac{876597147000000}{2458525710600000}v &=&-4\\ \end{array} \right. \)
Cliquer ici pour afficher la solution

Exercice


  1. \( \widehat{A}_{4, 5}=\begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -4 \\ -1 & 2 & -5 & -5 & -5 \\ -3 & 4 & 4 & \dfrac{6}{5} & 2 \\ -\dfrac{7}{4} & \dfrac{10}{3} & -\dfrac{23}{5} & \dfrac{23}{2} & 1 \\ \dfrac{16}{5} & -2 & 2 & 3 & \dfrac{8}{3}\end{pmatrix}\) \( \widehat{A}_{4, 6}=\begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -5 \\ -1 & 2 & -5 & -5 & 0 \\ -3 & 4 & 4 & \dfrac{6}{5} & 3 \\ -\dfrac{7}{4} & \dfrac{10}{3} & -\dfrac{23}{5} & \dfrac{23}{2} & -3 \\ \dfrac{16}{5} & -2 & 2 & 3 & 0\end{pmatrix}\)
  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 : \( L_{2}\leftarrow L_{2}-\left(-1\right)L_1\) , \( L_{3}\leftarrow L_{3}-\left(-3\right)L_1\) , \( L_{4}\leftarrow L_{4}-\left(-4\right)L_1\) , \( L_{5}\leftarrow L_{5}-\left(-\dfrac{7}{4}\right)L_1\) et \( L_{6}\leftarrow L_{6}-\left(\dfrac{16}{5}\right)L_1\)
  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{13}{4} & -\dfrac{21}{2} & -5 & -4 \\ 0 & 1 & -\dfrac{7}{4} & -\dfrac{31}{2} & -5 & -9 \\ 0 & 1 & \dfrac{55}{4} & -\dfrac{303}{10} & -12 & -10 \\ 0 & -1 & 15 & -\dfrac{202}{5} & -24 & -\dfrac{97}{5} \\ 0 & \dfrac{19}{12} & \dfrac{87}{80} & -\dfrac{55}{8} & -\dfrac{47}{4} & -6 \\ 0 & \dfrac{6}{5} & -\dfrac{42}{5} & \dfrac{183}{5} & 16 & \dfrac{232}{15}\end{pmatrix}\\ &=&1\times\det\begin{pmatrix}1 & -\dfrac{7}{4} & -\dfrac{31}{2} & -5 & -9 \\ 1 & \dfrac{55}{4} & -\dfrac{303}{10} & -12 & -10 \\ -1 & 15 & -\dfrac{202}{5} & -24 & -\dfrac{97}{5} \\ \dfrac{19}{12} & \dfrac{87}{80} & -\dfrac{55}{8} & -\dfrac{47}{4} & -6 \\ \dfrac{6}{5} & -\dfrac{42}{5} & \dfrac{183}{5} & 16 & \dfrac{232}{15}\end{pmatrix}\\ &=&-\dfrac{2048771425500}{81000000} \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{2048771425500}{81000000}\right)^{-1}{}^tCo\begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -5 & -4 \\ -1 & 2 & -5 & -5 & 0 & -5 \\ -3 & 4 & 4 & \dfrac{6}{5} & 3 & 2 \\ -4 & 3 & 2 & \dfrac{8}{5} & -4 & -\dfrac{17}{5} \\ -\dfrac{7}{4} & \dfrac{10}{3} & -\dfrac{23}{5} & \dfrac{23}{2} & -3 & 1 \\ \dfrac{16}{5} & -2 & 2 & 3 & 0 & \dfrac{8}{3}\end{pmatrix} =-\dfrac{81000000}{2048771425500}\begin{pmatrix}-\dfrac{9.9945339376452E+24}{7.4677718459475E+21} & -\dfrac{2.515923865492E+26}{3.7338859229737E+22} & -\dfrac{7.5620815068375E+25}{2.987108738379E+22} & \dfrac{1.1916406509821E+25}{2.987108738379E+22} & -\dfrac{8.2930436101674E+24}{9.95702912793E+21} & -\dfrac{1.1870569326231E+27}{9.95702912793E+22} \\ -\dfrac{5.8117021663416E+25}{1.659504854655E+22} & -\dfrac{4.7622228327831E+27}{9.95702912793E+23} & -\dfrac{6.1129022974615E+25}{9.95702912793E+21} & \dfrac{3.3679816975709E+25}{1.327603883724E+22} & -\dfrac{3.6691818286908E+25}{9.95702912793E+21} & -\dfrac{3.3189649026789E+26}{6.63801941862E+22} \\ \dfrac{2.2031437094963E+26}{2.4892572819825E+23} & -\dfrac{2.844341535733E+25}{4.1487621366375E+22} & -\dfrac{1.5271975111079E+26}{1.4935543691895E+23} & -\dfrac{1.5087554336581E+25}{4.1487621366375E+21} & \dfrac{2.9264040507957E+25}{1.2446286409912E+22} & -\dfrac{4.6891815520608E+26}{9.95702912793E+22} \\ \dfrac{4.9869216155587E+26}{1.8669429614869E+23} & -\dfrac{2.2035386716517E+26}{1.659504854655E+23} & \dfrac{2.5329022596599E+24}{2.4892572819825E+21} & -\dfrac{1.0040918757838E+27}{2.987108738379E+23} & \dfrac{2.6074306226825E+25}{2.4892572819825E+22} & -\dfrac{7.8156389130851E+26}{1.991405825586E+23} \\ \dfrac{2.6191843977165E+27}{6.2231432049562E+23} & -\dfrac{3.0370945388994E+30}{9.3347148074344E+26} & -\dfrac{1.566422065704E+27}{2.987108738379E+24} & -\dfrac{1.5524400735016E+27}{7.4677718459475E+23} & \dfrac{3.0401581300678E+26}{8.297524273275E+22} & -\dfrac{1.7018801451681E+27}{4.978514563965E+23} \\ -\dfrac{1.1673091567536E+27}{2.4892572819825E+23} & \dfrac{6.4786593247838E+27}{9.95702912793E+23} & -\dfrac{4.8401284541353E+25}{2.4892572819825E+22} & \dfrac{9.8736721990807E+25}{1.2446286409912E+22} & -\dfrac{7.8098455816375E+25}{1.659504854655E+22} & \dfrac{1.7959471864484E+27}{1.991405825586E+23}\end{pmatrix} =\begin{pmatrix}\dfrac{4878306000000}{92194714147500} & \dfrac{122801589000000}{460973570737500} & \dfrac{36910323000000}{368778856590000} & -\dfrac{5816367000000}{368778856590000} & \dfrac{4047813000000}{122926285530000} & \dfrac{579399399000000}{1229262855300000} \\ \dfrac{28366767000000}{204877142550000} & \dfrac{2324428569000000}{12292628553000000} & \dfrac{29836917000000}{122926285530000} & -\dfrac{16439031000000}{163901714040000} & \dfrac{17909181000000}{122926285530000} & \dfrac{161997813000000}{819508570200000} \\ -\dfrac{107534871000000}{3073157138250000} & \dfrac{13883157000000}{512192856375000} & \dfrac{74542113000000}{1843894282950000} & \dfrac{7364196000000}{51219285637500} & -\dfrac{14283702000000}{153657856912500} & \dfrac{228877731000000}{1229262855300000} \\ -\dfrac{243410346000000}{2304867853687500} & \dfrac{107554149000000}{2048771425500000} & -\dfrac{1236303000000}{30731571382500} & \dfrac{490094631000000}{3687788565900000} & -\dfrac{12726801000000}{307315713825000} & \dfrac{381479301000000}{2458525710600000} \\ -\dfrac{1278417087000000}{7682892845625000} & \dfrac{1482397939125000000}{1.1524339268437E+19} & \dfrac{764566533000000}{36877885659000000} & \dfrac{757741959000000}{9219471414750000} & -\dfrac{148389327000000}{1024385712750000} & \dfrac{830683269000000}{6146314276500000} \\ \dfrac{569760561000000}{3073157138250000} & -\dfrac{3162216753000000}{12292628553000000} & \dfrac{23624541000000}{307315713825000} & -\dfrac{48193137000000}{153657856912500} & \dfrac{38119653000000}{204877142550000} & -\dfrac{876597147000000}{2458525710600000}\end{pmatrix}\) . Précisément on a calculé \( A^{-1}_{6, 4}=B_{6, 4}= \left(-\dfrac{2048771425500}{81000000}\right)^{-1}Co(A)_{4, 6}= \left(-\dfrac{2048771425500}{81000000}\right)^{-1}\times(-1)^{4+6}\det\begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -5 \\ -1 & 2 & -5 & -5 & 0 \\ -3 & 4 & 4 & \dfrac{6}{5} & 3 \\ -\dfrac{7}{4} & \dfrac{10}{3} & -\dfrac{23}{5} & \dfrac{23}{2} & -3 \\ \dfrac{16}{5} & -2 & 2 & 3 & 0\end{pmatrix}=-\dfrac{48193137000000}{153657856912500}\) .
  6. On observe que \( B^{-1}=\left(A^{-1}\right)^{-1}=A\) . Trivialement, et sans calcul, \[ B^{-1}=\begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -5 & -4 \\ -1 & 2 & -5 & -5 & 0 & -5 \\ -3 & 4 & 4 & \dfrac{6}{5} & 3 & 2 \\ -4 & 3 & 2 & \dfrac{8}{5} & -4 & -\dfrac{17}{5} \\ -\dfrac{7}{4} & \dfrac{10}{3} & -\dfrac{23}{5} & \dfrac{23}{2} & -3 & 1 \\ \dfrac{16}{5} & -2 & 2 & 3 & 0 & \dfrac{8}{3}\end{pmatrix}\]
  7. Le système \( \left\{\begin{array}{*{7}{cr}} &x&-&y &+&\dfrac{13}{4}z &-&\dfrac{21}{2}t &-&5u &-&4v &=&-6\\ &-x&+&2y &-&5z &-&5t &&&-&5v &=&3\\ &-3x&+&4y &+&4z &+&\dfrac{6}{5}t &+&3u &+&2v &=&7\\ &-4x&+&3y &+&2z &+&\dfrac{8}{5}t &-&4u &-&\dfrac{17}{5}v &=&-4\\ &-\dfrac{7}{4}x&+&\dfrac{10}{3}y &-&\dfrac{23}{5}z &+&\dfrac{23}{2}t &-&3u &+&v &=&-7\\ &\dfrac{16}{5}x&-&2y &+&2z &+&3t &&&+&\dfrac{8}{3}v &=&-5\\ \end{array} \right.\) est équivalent à l'équation \( AX=a\) où la matrice \( A\) est celle de l'énoncé, \( X\) la matrice colonne des indéterminés et \( a=\begin{pmatrix}-6 \\ 3 \\ 7 \\ -4 \\ -7 \\ -5\end{pmatrix}\) de sorte que la solution est \( X=A^{-1}a=\begin{pmatrix}\dfrac{4878306000000}{92194714147500} & \dfrac{122801589000000}{460973570737500} & \dfrac{36910323000000}{368778856590000} & -\dfrac{5816367000000}{368778856590000} & \dfrac{4047813000000}{122926285530000} & \dfrac{579399399000000}{1229262855300000} \\ \dfrac{28366767000000}{204877142550000} & \dfrac{2324428569000000}{12292628553000000} & \dfrac{29836917000000}{122926285530000} & -\dfrac{16439031000000}{163901714040000} & \dfrac{17909181000000}{122926285530000} & \dfrac{161997813000000}{819508570200000} \\ -\dfrac{107534871000000}{3073157138250000} & \dfrac{13883157000000}{512192856375000} & \dfrac{74542113000000}{1843894282950000} & \dfrac{7364196000000}{51219285637500} & -\dfrac{14283702000000}{153657856912500} & \dfrac{228877731000000}{1229262855300000} \\ -\dfrac{243410346000000}{2304867853687500} & \dfrac{107554149000000}{2048771425500000} & -\dfrac{1236303000000}{30731571382500} & \dfrac{490094631000000}{3687788565900000} & -\dfrac{12726801000000}{307315713825000} & \dfrac{381479301000000}{2458525710600000} \\ -\dfrac{1278417087000000}{7682892845625000} & \dfrac{1482397939125000000}{1.1524339268437E+19} & \dfrac{764566533000000}{36877885659000000} & \dfrac{757741959000000}{9219471414750000} & -\dfrac{148389327000000}{1024385712750000} & \dfrac{830683269000000}{6146314276500000} \\ \dfrac{569760561000000}{3073157138250000} & -\dfrac{3162216753000000}{12292628553000000} & \dfrac{23624541000000}{307315713825000} & -\dfrac{48193137000000}{153657856912500} & \dfrac{38119653000000}{204877142550000} & -\dfrac{876597147000000}{2458525710600000}\end{pmatrix}\times\begin{pmatrix}-6 \\ 3 \\ 7 \\ -4 \\ -7 \\ -5\end{pmatrix}=\begin{pmatrix}-\dfrac{1.1718868427516E+87}{8.7338070452023E+86} \\ -\dfrac{8.7635435704775E+86}{5.1116931951801E+87} \\ -\dfrac{7.8934020951414E+87}{2.8079369358289E+88} \\ -\dfrac{2.0540490891423E+89}{4.0434291875937E+89} \\ \dfrac{2.9167532578958E+98}{1.8953574316845E+98} \\ \dfrac{3.4943064276823E+89}{8.9853981946526E+89}\end{pmatrix}\) . Ainsi \( x=\dfrac{1.1718868427516E+87}{8.7338070452023E+86}\) , \( y=\dfrac{8.7635435704775E+86}{5.1116931951801E+87}\) , \( z=\dfrac{7.8934020951414E+87}{2.8079369358289E+88}\) , \( t=\dfrac{2.0540490891423E+89}{4.0434291875937E+89}\) , \( u=\dfrac{2.9167532578958E+98}{1.8953574316845E+98}\) et \( v=\dfrac{3.4943064276823E+89}{8.9853981946526E+89}\)
  8. Le système \( \left\{\begin{array}{*{7}{cr}} &\dfrac{4878306000000}{92194714147500}x&+&\dfrac{122801589000000}{460973570737500}y &+&\dfrac{36910323000000}{368778856590000}z &-&\dfrac{5816367000000}{368778856590000}t &+&\dfrac{4047813000000}{122926285530000}u &+&\dfrac{579399399000000}{1229262855300000}v &=&-9\\ &\dfrac{28366767000000}{204877142550000}x&+&\dfrac{2324428569000000}{12292628553000000}y &+&\dfrac{29836917000000}{122926285530000}z &-&\dfrac{16439031000000}{163901714040000}t &+&\dfrac{17909181000000}{122926285530000}u &+&\dfrac{161997813000000}{819508570200000}v &=&0\\ &-\dfrac{107534871000000}{3073157138250000}x&+&\dfrac{13883157000000}{512192856375000}y &+&\dfrac{74542113000000}{1843894282950000}z &+&\dfrac{7364196000000}{51219285637500}t &-&\dfrac{14283702000000}{153657856912500}u &+&\dfrac{228877731000000}{1229262855300000}v &=&9\\ &-\dfrac{243410346000000}{2304867853687500}x&+&\dfrac{107554149000000}{2048771425500000}y &-&\dfrac{1236303000000}{30731571382500}z &+&\dfrac{490094631000000}{3687788565900000}t &-&\dfrac{12726801000000}{307315713825000}u &+&\dfrac{381479301000000}{2458525710600000}v &=&5\\ &-\dfrac{1278417087000000}{7682892845625000}x&+&\dfrac{1482397939125000000}{1.1524339268437E+19}y &+&\dfrac{764566533000000}{36877885659000000}z &+&\dfrac{757741959000000}{9219471414750000}t &-&\dfrac{148389327000000}{1024385712750000}u &+&\dfrac{830683269000000}{6146314276500000}v &=&-9\\ &\dfrac{569760561000000}{3073157138250000}x&-&\dfrac{3162216753000000}{12292628553000000}y &+&\dfrac{23624541000000}{307315713825000}z &-&\dfrac{48193137000000}{153657856912500}t &+&\dfrac{38119653000000}{204877142550000}u &-&\dfrac{876597147000000}{2458525710600000}v &=&-4\\ \end{array} \right.\) est équivalent à l'équation \( BX=b\) où la matrice \( B=A^{-1}\) déterminée précédement, \( X\) la matrice colonne des indéterminés et \( b=\begin{pmatrix}-9 \\ 0 \\ 9 \\ 5 \\ -9 \\ -4\end{pmatrix}\) de sorte que la solution est \( X=B^{-1}b=Ab=\begin{pmatrix}1 & -1 & \dfrac{13}{4} & -\dfrac{21}{2} & -5 & -4 \\ -1 & 2 & -5 & -5 & 0 & -5 \\ -3 & 4 & 4 & \dfrac{6}{5} & 3 & 2 \\ -4 & 3 & 2 & \dfrac{8}{5} & -4 & -\dfrac{17}{5} \\ -\dfrac{7}{4} & \dfrac{10}{3} & -\dfrac{23}{5} & \dfrac{23}{2} & -3 & 1 \\ \dfrac{16}{5} & -2 & 2 & 3 & 0 & \dfrac{8}{3}\end{pmatrix}\times\begin{pmatrix}-9 \\ 0 \\ 9 \\ 5 \\ -9 \\ -4\end{pmatrix}=\begin{pmatrix}\dfrac{115}{4} \\ -41 \\ 34 \\ \dfrac{558}{5} \\ \dfrac{1097}{20} \\ -\dfrac{97}{15}\end{pmatrix}\) . Ainsi \( x=\dfrac{115}{4}\) , \( y=41\) , \( z=34\) , \( t=\dfrac{558}{5}\) , \( u=\dfrac{1097}{20}\) et \( v=\dfrac{97}{15}\)