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MATRIKS

MATRIKS . INVERS MATRIKS ( dengan adjoint ). Adjoint. Definisi : Jika A sebarang matriks n x n dan C ij adalah kofaktor a ij , maka matriks dinamakan matriks kofaktor A Transpose dari matriks kofaktor adalah adjoint ( sering ditulis adj ( nama_matriks )

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MATRIKS

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  1. MATRIKS INVERS MATRIKS (denganadjoint)

  2. Adjoint • Definisi: • Jika A sebarangmatriks n x n danCijadalahkofaktoraij, makamatriks dinamakanmatrikskofaktor A • Transpose darimatrikskofaktoradalahadjoint (seringditulisadj(nama_matriks) • Transpose matrikskofaktor A adalahAdjoint A (adj(A))

  3. Adjoint • Contoh: • Carinilaikofaktor • C11 = (-1)1+1 (6*0 – 3*(-4)) = 12 • C12 = (-1)1+2 (1*0 – 3*2) = 6 • C13 = (-1)1+3 (1*(-4) – 6*2) = -16 • C21 = (-1)2+1 (2*0 – (-1)*(-4)) = 4 • C22 = (-1)2+2 (3*0 – (-1)*2) = 2 • C23 = (-1)2+3 (3*(-4)– 2*2) = 16 • C31 = (-1)3+1 (2*3 – (-1)*6) = 12 • C32 = (-1)3+2 (3*3 – (-1)*1) = -10 • C33 = (-1)3+3 (3*6 – 2*1) = 16 • MatriksKofaktor A • Transpose matrikskofaktor A adalahAdjoint A (adj(A))

  4. InversMatrikdenganAdjoint • Rumus:

  5. Contoh • Denganadjoint, carilahInversdari

  6. Contoh-penyelesaian • Carinilaikofaktor • C11 = (-1)1+1 (1*1 – 4*(-2)) = 9 • C12 = (-1)1+2 (0*1 – 4*2) = 8 • C13 = (-1)1+3 (0*(-2) – 1*2) = -2 • C21 = (-1)2+1 ((-1)*1 – 2*2) = 5 • C22 = (-1)2+2 (3*1 – 2*2) = -1 • C23 = (-1)2+3 (3*(-2) – (-1)*2) = 4 • C31 = (-1)3+1 ((-1)*4 – 2*1) = -6 • C32 = (-1)3+2 (3*4 – 2*0) = -12 • C33 = (-1)3+3 (3*1 – (-1)*0) = 3 • MatriksKofaktor A • Transpose matrikskofaktor A adalahAdjoint A (adj(A))

  7. Contoh-penyelesaian • CariDeterminannyadenganekspansikofaktorbarispertama: • det(A) = a11*c11+ a12*c12 a13*c13 = 3*9 + (-1)*8 + 2*(-2)  27 – 8 – 4 = 15

  8. METODE CRAMER

  9. Metode Cramer • untukmenyelesaikanpersamaan linier denganbantuandeterminan • SYARAT: nilaideterminan 0 (nol)

  10. Metode Cramer • jikaAx = badalahsebuahsistem linear n yang tidakdiketahuidandet(A)≠ 0 makapersamaantersebutmempunyaipenyelesaian yang unik • dimanaAjadalahmatrik yang didapatdenganmenggantikolomjdenganmatrik b

  11. LangkahMetode Cramer • Diketahui SPL: • Ubahterlebihdahuludalambentukmatriks • pisahkanmatriksuntukvariabeldankoefisiendisebelahkanansamadengan(=b)

  12. LangkahMetode Cramer • Diketahuimatriks A denganordo3x3, danmatrik b (matrikkolom) • Carideterminanmatriks A • Gantikolomdenganmatriks b • Gantikolompertamadenganmatriks b  • Gantikolomkeduadenganmatriks b  • Gantikolomketigadenganmatriks b 

  13. LangkahMetode Cramer • Carinilaideterminandarimatriksbaruhasilpenggantiankolomdenganmatriks b • Carinilai x1, x2dan x3denganrumusan:

  14. ContohSoal • Gunakanmetodecrameruntukmenyelesaikanpersoalandibawahini x1 + 2x3  = 6 -3x1 + 4x2 + 6x3 = 30 -x1 - 2x2 + 3x3  = 8

  15. PenyelesaianSoal • Bentukdalammatriks • Caridet(A), denganekspansibarispertama

  16. PenyelesaianSoal • Gantikolomdenganmatriks b • Carideterminanmasing-masing • denganekspansibarispertama

  17. PenyelesaianSoal • Gantikolomdenganmatriks b • Carideterminanmasing-masing • denganekspansibarispertama

  18. PenyelesaianSoal • Gantikolomdenganmatriks b • Carideterminanmasing-masing • denganekspansibarispertama

  19. PenyelesaianSoal • Carinilai x • Jadi, solusinya

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