A. Invers Matriks 2 × 2
Jika matriks \(A =
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\) maka invers matriks A adalah \(A^{-1} = \dfrac{1}{det \: A}
\begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}\)
Contoh:
Tentukan invers dari matriks \(A = \begin{pmatrix}3 & 5 \\2 & 4\end{pmatrix}\)
Jawab:
(1) Menentukan determinan matriks:
\(|A| = \begin{vmatrix}3 & 5 \\2 & 4\end{vmatrix} = 3 \:.\: 4 - 5 \:.\: 2 = 2\)
(2) Menentukan invers matriks:
\(A^{-1} = \dfrac{1}{2} \begin{pmatrix}4 & -5 \\-2 & 3\end{pmatrix} = \begin{pmatrix}2 & -2.5 \\-1 & 1.5\end{pmatrix}\)
B. Invers Matriks 3 × 3 (metode kofaktor)
\(A^{-1} = \dfrac{1}{det \: A} \: adj \: (A) \)
\(adj \: (A) = [kof \: (A)]^T\)
Menentukan Kofaktor
\(kof \: (A) =
\begin{pmatrix}
M_{11} & {\color {red} -}M_{12} & M_{13} \\
{\color {red} -}M_{21} & M_{22} & {\color {red} -}M_{23} \\
M_{31} & {\color {red} -}M_{32} & M_{33} \\
\end{pmatrix}\)
\(\begin{pmatrix} {\colorbox {cyan} a} & b & c \\d & {\color {red} e} & {\color {red} f} \\g & {\color {red} h} & {\color {red} i}\end{pmatrix} \rightarrow M_{11} = \begin{vmatrix} {\color {red} e} & {\color {red}f }\\ {\color {red}h} & {\color {red}i} \end{vmatrix}\)
\(\begin{pmatrix} a & {\colorbox {cyan} b} & c \\{\color {red} d} & e & {\color {red} f} \\{\color {red} g} & h & {\color {red} i}\end{pmatrix} \rightarrow M_{12} = \begin{vmatrix} {\color {red}d} & {\color {red}f} \\{\color {red} g} & {\color {red}i} \end{vmatrix}\)
\(\begin{pmatrix} a & b & {\colorbox {cyan} c} \\{\color {red} d} & {\color {red} e} & f \\{\color {red} g} & {\color {red} h} & i\end{pmatrix} \rightarrow M_{13} = \begin{vmatrix} {\color {red}d} & {\color {red}e} \\ {\color {red}g} & {\color {red}h} \end{vmatrix}\)
\(\begin{pmatrix} a & {\color {red} b} & {\color {red} c} \\{\colorbox {cyan} d} & e & f \\g & {\color {red} h} & {\color {red} i}\end{pmatrix} \rightarrow M_{21} = \begin{vmatrix} {\color {red} b} & {\color {red}c }\\ {\color {red}h} & {\color {red}i} \end{vmatrix}\)
\(\begin{pmatrix} {\color {red} a} & b & {\color {red} c} \\d & {\colorbox {cyan} e} & f \\{\color {red} g} & h & {\color {red} i}\end{pmatrix} \rightarrow M_{22} = \begin{vmatrix} {\color {red}a} & {\color {red}c} \\{\color {red} g} & {\color {red}i} \end{vmatrix}\)
\(\begin{pmatrix} {\color {red} a} & {\color {red} b} & c \\d & e & {\colorbox {cyan} f} \\{\color {red} g} & {\color {red} h} & i\end{pmatrix} \rightarrow M_{23} = \begin{vmatrix} {\color {red}a} & {\color {red}b} \\ {\color {red}g} & {\color {red}h} \end{vmatrix}\)
\(\begin{pmatrix} a & {\color {red} b} & {\color {red} c} \\d & {\color {red} e} & {\color {red} f} \\{\colorbox {cyan} g} & h & i\end{pmatrix} \rightarrow M_{31} = \begin{vmatrix} {\color {red} b} & {\color {red}c }\\ {\color {red}e} & {\color {red}f} \end{vmatrix}\)
\( \begin{pmatrix} {\color {red} a} & b & {\color {red} c} \\{\color {red} d} & e & {\color {red} f} \\g & {\colorbox {cyan} h} & i\end{pmatrix} \rightarrow M_{32} = \begin{vmatrix} {\color {red}a} & {\color {red}c} \\{\color {red} d} & {\color {red}f} \end{vmatrix}\)
\(\begin{pmatrix} {\color {red} a} & {\color {red} b} & c \\{\color {red} d} & {\color {red} e} & f \\g & h & {\colorbox {cyan} i}\end{pmatrix} \rightarrow M_{33} = \begin{vmatrix} {\color {red}a} & {\color {red}b} \\ {\color {red}d} & {\color {red}e} \end{vmatrix}\)
Contoh:
Tentukan invers dari matriks \(A = \begin{pmatrix}0 & -3 & -2 \\1 & -4 & -2 \\ -3 & 4 & 1 \end{pmatrix}\)
Jawab:
(1) Menentukan determinan (A)
\(|A| = 0 \: \begin{vmatrix} -4 & -2 \\4 & 1\end{vmatrix}- (-3) \:\begin{vmatrix}1 & -2 \\-3 & 1\end{vmatrix}+ (-2) \:\begin{vmatrix}1 & -4 \\-3 & 4\end{vmatrix}\)
\(|A| = 0 + 3 (1 - 6) - 2 (4 - 12)\)
\(|A| = 1\)
(2) Menentukan kofaktor (A)
\(kof \: (A) =
\begin{pmatrix}
\begin{vmatrix} -4 & -2 \\ 4 & 1 \end{vmatrix} &{\color {red} -}\begin{vmatrix} 1 & -2 \\ -3 & 1 \end{vmatrix} & \begin{vmatrix} 1 & -4 \\ -3 & 4 \end{vmatrix} \\
{\color {red} -}\begin{vmatrix} -3 & -2 \\ 4 & 1 \end{vmatrix} & \begin{vmatrix} 0 & -2 \\ -3 & 1 \end{vmatrix} & {\color {red} -}\begin{vmatrix} 0 & -3 \\ -3 & 4 \end{vmatrix} \\
\begin{vmatrix} -3 & -2 \\ -4 & -2 \end{vmatrix} & {\color {red} -}\begin{vmatrix} 0 & -2 \\ 1 & -2 \end{vmatrix} & \begin{vmatrix} 0 & -3 \\ 1 & -4 \end{vmatrix} \\
\end{pmatrix}
=
\begin{pmatrix}
4 & 5 & -8 \\
-5 & -6 & 9 \\
-2 & -2 & 3 \\
\end{pmatrix}
\)
(3) Menentukan Adjoin (A)
\(Adj \: (A) = [kof \: (A)]^T =
\begin{pmatrix}
4 & -5 & -2 \\
5 & -6 & -2 \\
-8 & 9 & 3 \\
\end{pmatrix}
\)
(4) Menentukan \(A^{-1}\)
\(A^{-1} = \dfrac{1}{det \: A} \: Adj \: (A) = \dfrac{1}{1}
\begin{pmatrix}
4 & -5 & -2 \\
5 & -6 & -2 \\
-8 & 9 & 3 \\
\end{pmatrix}
=
\begin{pmatrix}
4 & -5 & -2 \\
5 & -6 & -2 \\
-8 & 9 & 3 \\
\end{pmatrix}
\)
C. Sifat-sifat Invers Matriks
-
- \(A \:.\: A^{-1} = A^{-1} \:.\: A = I\)
- \((A \:.\: B)^{-1} = B^{-1} \:.\: A^{-1}\)