\(\color{blue}\textbf{A} = \begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22} \end{bmatrix}\)
Invers matriks \(\textbf{A}\) ditulis dengan \(\textbf{A}^{-1}\)
$$\color{blue}\textbf{A}^{-1} = \dfrac{1}{\textbf{determinan A}} \cdot \textbf{adjoin A}$$
$$\color{blue}\textbf{A}^{-1} = \dfrac{1}{a_{11}\cdot a_{22}\:-\:a_{12} \cdot a_{21}} \cdot \begin{bmatrix}a_{22} & -a_{12} \\-a_{21} & a_{11} \end{bmatrix}$$
Hasil perkalian matriks \(\textbf{A}\) dengan \(\textbf{A}^{-1}\) akan menghasilkan matriks identitas \(\textbf{I}\)
\(\color{blue}\textbf{A}\cdot \textbf{A}^{-1} = \textbf{I}\)
\(\color{blue}\textbf{A}\cdot \textbf{A}^{-1} = \begin{bmatrix}1 & 0 \\0 &1 \end{bmatrix}\)
Contoh 1
Tentukan invers matriks A berikut:
\(\textbf{A} = \begin{bmatrix}1 & 3 \\2& 8 \end{bmatrix}\)
Penyelesaian
\(\textbf{A}^{-1} = \dfrac{1}{1\cdot 8\:-\:3\cdot 2} \cdot \begin{bmatrix}8 & -3 \\-2 & 1 \end{bmatrix}\)
\(\textbf{A}^{-1} = \dfrac{1}{2} \cdot \begin{bmatrix}8 & -3 \\-2 & 1 \end{bmatrix}\)
\(\textbf{A}^{-1} = \begin{bmatrix}4 & -\frac{3}{2} \\-1 & \frac{1}{2} \end{bmatrix}\)
Contoh 2
Tentukan invers matriks B berikut:
\(\textbf{B} = \begin{bmatrix}-2 & 3 \\ 3& -6 \end{bmatrix}\)
Penyelesaian
\(\textbf{B}^{-1} = \dfrac{1}{-2\cdot (-6)\:-\:3\cdot 3} \cdot \begin{bmatrix}-6 & -3 \\-3 & -2 \end{bmatrix}\)
\(\textbf{B}^{-1} = \dfrac{1}{12\:-\:9} \cdot \begin{bmatrix}-6 & -3 \\-3 & -2 \end{bmatrix}\)
\(\textbf{B}^{-1} = \dfrac{1}{3} \cdot \begin{bmatrix}-6 & -3 \\-3 & -2 \end{bmatrix}\)
\(\textbf{B}^{-1} = \begin{bmatrix}-2 & -1 \\-1 & -\frac{2}{3} \end{bmatrix}\)