”[ rac{1}{{12}}left[
Answer is Right!
Answer is Wrong!
The correct answer is $\boxed{\frac{1}{14}\left[\begin{array}{cc}3-2i&-i\\i&3+2i\end{array}\right]}$.
To find the inverse of a 2×2 matrix, we can use the formula $A^{-1}=\frac{1}{|A|}\left[\begin{array}{cc}a_{2,2}&-a_{1,2}\\a_{2,1}&a_{1,1}\end{array}\right]$, where $|A|$ is the determinant of $A$.
In this case, $A=\left[\begin{array}{cc}3+2i&i\-i&3-2i\end{array}\right]$. Therefore,
\begin{align}
|A|&=(3+2i)(3-2i)-i(-i) \\
&=9-4i+i^2 \\
&=12
\end{align}
Therefore, the inverse of $A$ is $\frac{1}{|A|}\left[\begin{array}{cc}3-2i&-i\\i&3+2i\end{array}\right]=\frac{1}{12}\left[\begin{array}{cc}3-2i&-i\\i&3+2i\end{array}\right]$.