Invers Matriks matriks $2\times 2$

 Invers Matriks Ordo $2\times 2$

Jika Matriks $A=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)$, maka invers matriks A ditulis $A^{-1}$.

$A^{-1}=\frac{1}{|A|}\left(\begin{matrix}d&-b\\-c&a\end{matrix}\right)$

Contoh 7

Diketahui matriks $A=\left(\begin{matrix}6&4\\8&5\end{matrix}\right)$, tentukan invers dari matriks A.

Jawab

 $|A|=(6\times 5)-(4\times 8)$

     $=30-32$

     $=-2$

$A^{-1}=\frac{1}{-2}\left(\begin{matrix}5&-4\\-8&6\end{matrix}\right)$

$A^{-1}=\left(\begin{matrix}-\frac{5}{2}&\frac{4}{2}\\\frac{8}{2}&-\frac{6}{2}\end{matrix}\right)$

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

jadi invers dari matriks A yaitu $A^{-1}=\left(\begin{matrix}-\frac{5}{2}&2\\4&-3\end{matrix}\right)$

Latihan 7

1.  Diketahui matriks $A=\left(\begin{matrix}9&4\\7&3\end{matrix}\right)$, tentukan invers matriks A.

2.  Diketahui matriks $P=\left(\begin{matrix}0&3\\\frac{1}{3}&2\end{matrix}\right)$, tentukan invers matriks P.

Contoh 8

Diketahui matriks $R=\left(\begin{matrix}\text{cos }\alpha&-\text{sin }\alpha\\\text{sin }\alpha&\text{cos }\alpha\end{matrix}\right)$, tentukan invers dari matriks R jika $\alpha=30^{\circ}$.

Jawab

$R=\left(\begin{matrix}\text{cos }30^{\circ}&-\text{sin }30^\circ\\\text{sin }30^\circ&\text{cos }30^\circ\end{matrix}\right)$

$R=\left(\begin{matrix}\frac{1}{2}\sqrt{3}&-\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}\sqrt{3}\end{matrix}\right)$

Determinan matriks R

$|R|=\left(\frac{1}{2}\sqrt{3}\right)\left(\frac{1}{2}\sqrt{3}\right)-\left(-\frac{1}{2}\right)\left(\frac{1}{2}\right)$

     $=\frac{3}{4}+\frac{1}{4}$

     $=\frac{4}{4}=1$

Invers matriks R

$R^{-1}=\frac{1}{1}\left(\begin{matrix}\frac{1}{2}\sqrt{3}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\sqrt{3}\end{matrix}\right)$

            $=\left(\begin{matrix}\frac{1}{2}\sqrt{3}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\sqrt{3}\end{matrix}\right)$

Jadi, invers matriks R adalah $=\left(\begin{matrix}\frac{1}{2}\sqrt{3}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\sqrt{3}\end{matrix}\right)$

Latihan 8

1.   Diketahui matriks $R=\left(\begin{matrix}\text{cos }\alpha&-\text{sin }\alpha\\\text{sin }\alpha&\text{cos }\alpha\end{matrix}\right)$, tentukan invers dari matriks R jika $\alpha=45^{\circ}$.

2.  Diketahui matriks $R=\left(\begin{matrix}\text{cos }\alpha&-\text{sin }\alpha\\\text{sin }\alpha&\text{cos }\alpha\end{matrix}\right)$, tentukan invers dari matriks R jika $\alpha$ sudut istimewa lebih besar dari $90^\circ$.