Invers Matriks $3\times 3$

Diketahui Matriks $A=\left(\begin{matrix}a&b&c\\d&e&f\\g&h&i\end{matrix}\right)$, invers dari matriks A adalah $A^{-1}=\frac{1}{|A|}. \text{Adj. A}$

Cara mencari Adj. A

1.  Menentukan Minor masing-masing entry.
    -  Minor entry a (tutup sebaris  dan sekolom dengan a) 
       $\text{Mi}_a=\left|\begin{matrix}e&f\\h&i\end{matrix}\right|=e\times i-f\times h$
     - Minor entry b (tutup sebaris dan sekolom dengan b)
        $\text{Mi}_b=\left|\begin{matrix}d&f\\g&i\end{matrix}\right|=d\times i-f\times g$

     - Minor entry c 
        $\text{Mi}_c=\left|\begin{matrix}d&e\\g&h\end{matrix}\right|=d\times h-e\times g$

    -  Minor entry d
       $\text{Mi}_d=\left|\begin{matrix}b&c\\h&i\end{matrix}\right|=b\times i-c\times h$

    -  Minor entry e

       $\text{Mi}_e=\left|\begin{matrix}a&c\\g&i\end{matrix}\right|=a\times i-c\times g$

    -  Minor entry f

       $\text{Mi}_f=\left|\begin{matrix}a&b\\g&h\end{matrix}\right|=a\times h-b\times g$

    -  Minor entry g

       $\text{Mi}_g=\left|\begin{matrix}b&c\\e&f\end{matrix}\right|=b\times f-c\times e$

    -  Minor entry h

       $\text{Mi}_h=\left|\begin{matrix}a&c\\d&f\end{matrix}\right|=a\times f-c\times d$

    -  Minor entry i

       $\text{Mi}_i=\left|\begin{matrix}a&b\\d&e\end{matrix}\right|=a\times e-b\times d$

2.  Menentukan kofaktor

      $Kofaktor A=\left(\begin{matrix}\text{Mi}_a&-(\text{Mi}_b)&\text{Mi}_c\\-(\text{Mi}_d)&\text{Mi}_e&-(\text{Mi}_f)\\\text{Mi}_g&-(\text{Mi}_h)&\text{Mi}_i\end{matrix}\right)$

3.  Menentukan Adjoin (Adj)

     Adjoin merupakan matriks hasil dari transpose matriks kofaktor.

     $\text{Adj}. A=\left(\begin{matrix}\text{Mi}_a&-(\text{Mi}_d)&\text{Mi}_g \\-(\text{Mi}_b&\text{Mi}_e&-(\text{Mi}_h)\\ \text{Mi}_c&-(\text{Mi}_f)&\text{Mi}_i\end{matrix}\right)$  

Contoh

Diketahui matriks $A=\left(\begin{matrix}2&1&3\\-2&5&8\\7&3&1\end{matrix}\right)$, tentukan invers dari matriks A.

Jawab;

$\left(\begin{matrix}2&1&3\\-2&5&8\\7&3&1\end{matrix}\right)\begin{array}\text{2}&1\\-2&5\\7&3\end{array}$

Determinan matriks A

$|A|=(2\times 5\times 1+1\times 8\times 7+3\times (-2)\times3)-(1\times (-2)\times 1+2\times 8\times 3+3\times 5\times 7)$

    $=(10+56-18)-(-2+48+105)$

    $=(48)-(151)=-103$

Video Determinan Matriks A

Minor Matriks  A

$\text{Minor Matriks A}=\left(\begin{matrix}5\times 1-8\times 3&(-2)\times1-8\times 7&(-2)\times 3-5\times 7\\1\times 1-3\times 3&2\times 1-3\times 7&2\times 3-1\times 7\\1\times 8-3\times 5&2\times 8-3\times (-2)&2\times 5-1\times (-2)\end{matrix}\right)$ 

                $=\left(\begin{matrix}5-24&-2-56&-6-35\\1-9&2-21&6-7\\8-15&16+6&10+2\end{matrix}\right)$

                $=\left(\begin{matrix}-19&-58&-41\\-8&-19&-1\\-7&22&12\end{matrix}\right)$

Video Menentukan Minor Matriks

Kofaktor Matriks A

$\text{Koofaktor Matriks A }=\left(\begin{matrix}-19&-(-58)&-41\\-(-8)&-19&-(-1)\\-7&-(22)&12\end{matrix}\right)$

                   $=\left(\begin{matrix}-19&58&41\\8&-19&1\\-7&-22&12\end{matrix}\right)$

Adjoin Matriks A

$\text{Adj. A }=\left(\begin{matrix}-19&8&-7\\58&-19&-22\\41&1&12\end{matrix}\right)$

Invers Matriks A

$A^{-1}=\frac{1}{103}\left(\begin{matrix}-19&8&-7\\58&-19&-22\\41&1&12\end{matrix}\right)$

          

    $=\left(\begin{matrix}\frac{-19}{103}&\frac{8}{103}&\frac{-7}{103}\\\frac{58}{103}&\frac{-19}{103}&\frac{-22}{103}\\\frac{41}{103}&\frac{1}{103}&\frac{12}{103}\end{matrix}\right)$

Jadi invers dari matriks A yaitu $A^{-1}=\left(\begin{matrix}\frac{-19}{103}&\frac{8}{103}&\frac{-7}{103}\\\frac{58}{103}&\frac{-19}{103}&\frac{-22}{103}\\\frac{41}{103}&\frac{1}{103}&\frac{12}{103}\end{matrix}\right)$

Latihan 

1.  Diketahui matriks $B=\left(\begin{matrix}a&3&2\\-4&1&b\\c&5&6\end{matrix}\right)$, tentukan invers dari matriks B. (ganti a,b c dengan bilangan bulat).