# Invers Matriks

## Konsep Dasar

#### 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)

$$kof \: (A) = \begin{pmatrix} \begin{vmatrix} -4 & -2 \\ 4 & 1 \end{vmatrix} &{\color {red} -}\begin{vmatrix} 1 & -2 \\ -3 & 1 \end{vmatrix} & \begin{vmatrix} 1 & -4 \\ -3 & 4 \end{vmatrix} \\ {\color {red} -}\begin{vmatrix} -3 & -2 \\ 4 & 1 \end{vmatrix} & \begin{vmatrix} 0 & -2 \\ -3 & 1 \end{vmatrix} & {\color {red} -}\begin{vmatrix} 0 & -3 \\ -3 & 4 \end{vmatrix} \\ \begin{vmatrix} -3 & -2 \\ -4 & -2 \end{vmatrix} & {\color {red} -}\begin{vmatrix} 0 & -2 \\ 1 & -2 \end{vmatrix} & \begin{vmatrix} 0 & -3 \\ 1 & -4 \end{vmatrix} \\ \end{pmatrix} = \begin{pmatrix} 4 & 5 & -8 \\ -5 & -6 & 9 \\ -2 & -2 & 3 \\ \end{pmatrix}$$

$$Adj \: (A) = [kof \: (A)]^T = \begin{pmatrix} 4 & -5 & -2 \\ 5 & -6 & -2 \\ -8 & 9 & 3 \\ \end{pmatrix}$$

(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

• $$A \:.\: A^{-1} = A^{-1} \:.\: A = I$$
• $$(A \:.\: B)^{-1} = B^{-1} \:.\: A^{-1}$$