Invers matriks

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A. Invers Matriks 2 × 2

Jika matriks $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ maka invers matriks A adalah $A^{- 1} = \frac{1}{d e t A} (\begin{matrix} d & - b \\ - c & a \end{matrix})$

Contoh:

Tentukan invers dari matriks $A = (\begin{matrix} 3 & 5 \\ 2 & 4 \end{matrix})$

Jawab:

(1) Menentukan determinan matriks:

$| A | = | \begin{matrix} 3 & 5 \\ 2 & 4 \end{matrix} | = 3 . 4 - 5 . 2 = 2$

(2) Menentukan invers matriks:

$A^{- 1} = \frac{1}{2} (\begin{matrix} 4 & - 5 \\ - 2 & 3 \end{matrix}) = (\begin{matrix} 2 & - 2.5 \\ - 1 & 1.5 \end{matrix})$

B. Invers Matriks 3 × 3 (metode kofaktor)

$A^{- 1} = \frac{1}{d e t A} a d j (A)$

$a d j (A) = [k o f (A)]^{T}$

Menentukan Kofaktor

$k o f (A) = (\begin{matrix} M_{11} & - M_{12} & M_{13} \\ - M_{21} & M_{22} & - M_{23} \\ M_{31} & - M_{32} & M_{33} \end{matrix})$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{11} = | \begin{matrix} e & f \\ h & i \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{12} = | \begin{matrix} d & f \\ g & i \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{13} = | \begin{matrix} d & e \\ g & h \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{21} = | \begin{matrix} b & c \\ h & i \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{22} = | \begin{matrix} a & c \\ g & i \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{23} = | \begin{matrix} a & b \\ g & h \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{31} = | \begin{matrix} b & c \\ e & f \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{32} = | \begin{matrix} a & c \\ d & f \end{matrix} |$

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) \to M_{33} = | \begin{matrix} a & b \\ d & e \end{matrix} |$

Contoh:

Tentukan invers dari matriks $A = (\begin{matrix} 0 & - 3 & - 2 \\ 1 & - 4 & - 2 \\ - 3 & 4 & 1 \end{matrix})$

Jawab:

(1) Menentukan determinan (A)

$| A | = 0 | \begin{matrix} - 4 & - 2 \\ 4 & 1 \end{matrix} | - (- 3) | \begin{matrix} 1 & - 2 \\ - 3 & 1 \end{matrix} | + (- 2) | \begin{matrix} 1 & - 4 \\ - 3 & 4 \end{matrix} |$

$| A | = 0 + 3 (1 - 6) - 2 (4 - 12)$

$| A | = 1$

(2) Menentukan kofaktor (A)

$k o f (A) = (\begin{matrix} | \begin{matrix} - 4 & - 2 \\ 4 & 1 \end{matrix} | & - | \begin{matrix} 1 & - 2 \\ - 3 & 1 \end{matrix} | & | \begin{matrix} 1 & - 4 \\ - 3 & 4 \end{matrix} | \\ - | \begin{matrix} - 3 & - 2 \\ 4 & 1 \end{matrix} | & | \begin{matrix} 0 & - 2 \\ - 3 & 1 \end{matrix} | & - | \begin{matrix} 0 & - 3 \\ - 3 & 4 \end{matrix} | \\ | \begin{matrix} - 3 & - 2 \\ - 4 & - 2 \end{matrix} | & - | \begin{matrix} 0 & - 2 \\ 1 & - 2 \end{matrix} | & | \begin{matrix} 0 & - 3 \\ 1 & - 4 \end{matrix} | \end{matrix}) = (\begin{matrix} 4 & 5 & - 8 \\ - 5 & - 6 & 9 \\ - 2 & - 2 & 3 \end{matrix})$

(3) Menentukan Adjoin (A)

$A d j (A) = [k o f (A)]^{T} = (\begin{matrix} 4 & - 5 & - 2 \\ 5 & - 6 & - 2 \\ - 8 & 9 & 3 \end{matrix})$

(4) Menentukan $A^{- 1}$

$A^{- 1} = \frac{1}{d e t A} A d j (A) = \frac{1}{1} (\begin{matrix} 4 & - 5 & - 2 \\ 5 & - 6 & - 2 \\ - 8 & 9 & 3 \end{matrix}) = (\begin{matrix} 4 & - 5 & - 2 \\ 5 & - 6 & - 2 \\ - 8 & 9 & 3 \end{matrix})$

C. Sifat-sifat Invers Matriks

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

SOAL LATIHAN

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