Adjoint dan Invers Matriks 5x5 (Metode Kofaktor)

(\mathrm{adj}\,A)_{ij} = (-1)^{i+j} M_{ji}

Jika ingin ditulis dengan penjelasan minor:

M_{ji} = \det(A_{ji})

Sehingga bisa juga ditulis:

(\mathrm{adj}\,A)_{ij} = (-1)^{i+j} \det(A_{ji})

Ini adalah rumus elemen ke-((i,j)) dari adjoint matriks.

A = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 7 & 8 & 9 & 1 & 2 \\ 3 & 4 & 5 & 6 & 7 \end{pmatrix}

1. Kofaktor¶

Rumus kofaktor:

C_{ij} = (-1)^{i+j} \det(A_{ij})

Contoh:

Coret baris 1 kolom 1:

A_{11} = \begin{pmatrix} 7 & 8 & 9 & 1 \\ 3 & 4 & 5 & 6 \\ 8 & 9 & 1 & 2 \\ 4 & 5 & 6 & 7 \end{pmatrix}

C_{11} = (+1)\det(A_{11})

A_{12} = \begin{pmatrix} 6 & 8 & 9 & 1 \\ 2 & 4 & 5 & 6 \\ 7 & 9 & 1 & 2 \\ 3 & 5 & 6 & 7 \end{pmatrix}

C_{12} = (-1)\det(A_{12})

Proses ini dilakukan untuk semua elemen sampai C55.

C = \begin{pmatrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} \\ C_{21} & C_{22} & C_{23} & C_{24} & C_{25} \\ C_{31} & C_{32} & C_{33} & C_{34} & C_{35} \\ C_{41} & C_{42} & C_{43} & C_{44} & C_{45} \\ C_{51} & C_{52} & C_{53} & C_{54} & C_{55} \end{pmatrix}

Transpose matriks kofaktor:

\text{adj}(A) = \begin{pmatrix} C_{11} & C_{21} & C_{31} & C_{41} & C_{51} \\ C_{12} & C_{22} & C_{32} & C_{42} & C_{52} \\ C_{13} & C_{23} & C_{33} & C_{43} & C_{53} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{54} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} \end{pmatrix}

Jika determinan tidak nol:

A^{-1} = \frac{1}{\det(A)} \text{adj}(A)

A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} C_{11} & C_{21} & C_{31} & C_{41} & C_{51} \\ C_{12} & C_{22} & C_{32} & C_{42} & C_{52} \\ C_{13} & C_{23} & C_{33} & C_{43} & C_{53} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{54} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} \end{pmatrix}