b'\n_START_ARTICLE_\nApgriezt\xc4\x81 matrica\n_START_PARAGRAPH_\nKvadr\xc4\x81tiskas, nesingul\xc4\x81ras matricas A apgriezt\xc4\x81 matrica jeb invers\xc4\x81 matrica ir t\xc4\x81da matrica, kuru reizinot ar matricu A, ieg\xc5\xabst vien\xc4\xabbas matricu:_NEWLINE_A^{-1} \\cdot A = A \\cdot A^{-1} = E\n_START_SECTION_\nIzmantojot minorus\n_START_PARAGRAPH_\nA^{-1} = _NEWLINE_\\begin{pmatrix}_NEWLINE_\\frac{A_{11}}{d} & \\frac{A_{21}}{d} & \\cdots & \\frac{A_{n1}}{d} \\\\_NEWLINE_\\frac{A_{12}}{d} & \\frac{A_{22}}{d} & \\cdots & \\frac{A_{n2}}{d} \\\\_NEWLINE_\\cdots & \\cdots & \\cdots & \\cdots \\\\_NEWLINE_\\frac{A_{1n}}{d} & \\frac{A_{2n}}{d} & \\cdots & \\frac{A_{nn}}{d}_NEWLINE_\\end{pmatrix}_NEWLINE_kur d ir matricas A determinants un A_{ij} ir algebriskais papildin\xc4\x81jums matricas minoram M_{ij} = \\left | a_{ij}\\right |:_NEWLINE_A_{ij} = (-1)^{i+j} \\cdot M_{ij}._NEWLINE_\xc5\xa0\xc4\xab formula r\xc4\x81da, ka ieg\xc5\xabt apgriezto matricu (jeb, citiem v\xc4\x81rdiem, invert\xc4\x93t doto matricu) ir iesp\xc4\x93jams tikai tad, ja matricas determinants nav vien\xc4\x81ds nullei. \xc5\xa0o krit\xc4\x93riju izmanto matricas apgrie\xc5\xbeam\xc4\xabbas p\xc4\x81rbaudei.\n_START_SECTION_\nIzmantojot Gausa metodi\n_START_PARAGRAPH_\nPie\xc5\x86emam, ka esam atradu\xc5\xa1i matricas T_1 l\xc4\xabdz T_n, kuras sec\xc4\xabgi piereizinot matricai A, ieg\xc5\xabstam vien\xc4\xabbas matricu E:_NEWLINE_T_n \\cdot \\ldots \\cdot\\ T_2 \\cdot T_1 \\cdot A = E_NEWLINE_T\xc4\x81d\xc4\x81 gad\xc4\xabjum\xc4\x81 viegli redz\xc4\x93t, ka varam izteikt apgriezto matricu k\xc4\x81 T_i matricu reizin\xc4\x81jumu:_NEWLINE_A^{-1} = T_n \\cdot \\ldots \\cdot T_2 \\cdot T_1 (\\cdot E)_NEWLINE_L\xc4\xabdz ar to apgriezto matricu ir iesp\xc4\x93jams mekl\xc4\x93t k\xc4\x81 matricu T_i reizin\xc4\x81jumu, ko savuk\xc4\x81rt var atrast, p\xc4\x81rveidojot matricu A par vien\xc4\xabbas matricu E \xe2\x80\x94 p\xc4\x93c Gausa metodes._NEWLINE_\xc5\xa0im nol\xc5\xabkam saraksta kop\xc4\x81 matricu A un vien\xc4\xabbas matricu E un ar matricu A veic element\xc4\x81ros matricu p\xc4\x81rveidojumus, lai p\xc4\x81rveidotu to par E. Tos pa\xc5\xa1us p\xc4\x81rveidojumus paral\xc4\x93li veic ar\xc4\xab ar matricu E. Br\xc4\xabd\xc4\xab, kad kreisaj\xc4\x81 pus\xc4\x93 esam ieguvu\xc5\xa1i vien\xc4\xabbas matricu, labaj\xc4\x81 pus\xc4\x93 ir redzama matricas A apgriezt\xc4\x81:_NEWLINE_(A | E) \\sim \\ldots \\sim (E | A^{-1})\n_START_SECTION_\nPiem\xc4\x93rs\n_START_PARAGRAPH_\nDota matrica _NEWLINE_A = \\begin{pmatrix}_NEWLINE_1 & 2 & 3 \\\\_NEWLINE_0 & 1 & 1 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{pmatrix}_NEWLINE__NEWLINE_\\left (_NEWLINE_\\begin{matrix}_NEWLINE_1 & 2 & 3 \\\\_NEWLINE_0 & 1 & 1 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{matrix}_NEWLINE_\\right |_NEWLINE_\\left ._NEWLINE_\\begin{matrix}_NEWLINE_1 & 0 & 0 \\\\_NEWLINE_0 & 1 & 0 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{matrix}_NEWLINE_\\right )_NEWLINE__NEWLINE_(1)\\sim_NEWLINE__NEWLINE_\\left (_NEWLINE_\\begin{matrix}_NEWLINE_1 & 2 & 0 \\\\_NEWLINE_0 & 1 & 0 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{matrix}_NEWLINE_\\right |_NEWLINE_\\left ._NEWLINE_\\begin{matrix}_NEWLINE_1 & 0 & -3 \\\\_NEWLINE_0 & 1 & -1 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{matrix}_NEWLINE_\\right )_NEWLINE__NEWLINE_(2)\\sim_NEWLINE__NEWLINE_\\left (_NEWLINE_\\begin{matrix}_NEWLINE_1 & 0 & 0 \\\\_NEWLINE_0 & 1 & 0 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{matrix}_NEWLINE_\\right |_NEWLINE_\\left ._NEWLINE_\\begin{matrix}_NEWLINE_1 & -2 & -1 \\\\_NEWLINE_0 & 1 & -1 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{matrix}_NEWLINE_\\right )_NEWLINE_(1) p\xc4\x81rveidojum\xc4\x81 3. rindi\xc5\x86a tika pareizin\xc4\x81ta ar -1 un pieskait\xc4\xabta 2. rindi\xc5\x86ai; 3. rindi\xc5\x86a tika pareizin\xc4\x81ta ar -3 un pieskait\xc4\xabta 1. rindi\xc5\x86ai._NEWLINE__NEWLINE_(2) p\xc4\x81rveidojum\xc4\x81 2. rindi\xc5\x86a tika pareizin\xc4\x81ta ar -2 un pieskait\xc4\xabta 1. rindi\xc5\x86ai._NEWLINE_Rezult\xc4\x81t\xc4\x81 esam ieguvu\xc5\xa1i apgriezto matricu _NEWLINE_A^{-1} = \\begin{pmatrix}_NEWLINE_1 & -2 & -1 \\\\_NEWLINE_0 & 1 & -1 \\\\_NEWLINE_0 & 0 & 1_NEWLINE_\\end{pmatrix}'