Diketahui \(\textbf{A} = \begin{bmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22}& a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\). Invers dari matriks \(\textbf{A}\) ditulis \(\textbf{A}^{-1}\).
$$\bbox[yellow, 5px, border: 2px solid red] {\text{A}^{-1} = \dfrac{1}{\text{determinan A}} \cdot \text{adjoint A}}$$
\(\color{blue} \text{Adjoint A} = \text{transpose dari matriks kofaktor A}\)
\(\color{blue} \text{Kofaktor A} = \begin{bmatrix}+ M_{11} & -M_{12} & +M_{13} \\-M_{21} &+M_{22}& -M_{23} \\ +M_{31} & -M_{32} & +M_{33} \end{bmatrix}\).
\(M_{11}\) adalah determinan matriks dengan menutup baris 1 dan kolom 1 pada matriks A
\(M_{11} =\begin{vmatrix}a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}\)
\(M_{12}\) adalah determinan matriks dengan menutup baris 1 dan kolom 2 pada matriks A
\(M_{12} = \begin{vmatrix}a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}\)
Dan seterusnya…
CONTOH SOAL
Tentukan invers matriks \(\textbf{A} = \begin{bmatrix}1 & 2 & 3 \\4& 5& 6 \\ 7 & 8 & 0 \end{bmatrix}\)
Penyelesaian:
Langkah 1: Menentukan determinan matriks A
\(|A| = +1\cdot \begin{vmatrix}5 & 6 \\ 8 & 0\end{vmatrix} \:-\:2 \cdot \begin{vmatrix}4 & 6 \\7 & 0\end{vmatrix} + 3 \cdot \begin{vmatrix}4 & 5 \\ 7 & 8\end{vmatrix}\)
\(|A| = +1\cdot (-48) \:-\:2 \cdot (-42)+ 3 \cdot (-3)\)
\(|A| = -48 + 84\:-\:9 = 27\)
Langkah 2: Menentukan matriks kofaktor A
\(\text{Kofaktor A} = \begin{bmatrix}+ \begin{vmatrix}5 & 6 \\ 8 & 0\end{vmatrix} & -\begin{vmatrix}4 & 6 \\ 7 & 0\end{vmatrix} & +\begin{vmatrix}4 & 5 \\ 7& 8\end{vmatrix} \\-\begin{vmatrix}2 & 3 \\ 8 & 0\end{vmatrix} &+\begin{vmatrix}1& 3 \\7 & 0\end{vmatrix}& -\begin{vmatrix}1 & 2 \\ 7 & 8\end{vmatrix} \\ +\begin{vmatrix}2 & 3\\ 5 & 6\end{vmatrix} & -\begin{vmatrix}1 & 3 \\ 4 & 6\end{vmatrix} & +\begin{vmatrix}1 & 2 \\ 4 & 5\end{vmatrix} \end{bmatrix}\).
\(\text{Kofaktor A} = \begin{bmatrix}-48& 42 & -3 \\24 &-21& 6 \\ -3 & 6 &-3 \end{bmatrix}\).
Langkah 3: Menentukan matriks adjoint A
\(\text{Adjoint A} = \text{transpose dari matriks kofaktor A}\)
\(\text{Adjoint A} = \begin{bmatrix}-48& 24 & -3 \\42 &-21& 6 \\ -3 & 6 &-3 \end{bmatrix}\).
Langkah 4: Menulis invers matriks A
\(\text{A}^{-1} = \dfrac{1}{\text{determinan A}} \cdot \text{adjoint A}\)
\(\text{A}^{-1} = \dfrac{1}{27}\begin{bmatrix}-48& 24 & -3 \\42 &-21& 6 \\ -3 & 6 &-3 \end{bmatrix}\).