11. Find the inverse of the following matrix, if it exists.
\[
\left[\begin{array}{ll}
8 & 3 \\
5 & 2
\end{array}\right]
\]

Step 1 ：We are given the matrix \(A = \left[\begin{array}{ll} 8 & 3 \\ 5 & 2 \end{array}\right]\)

Step 2 ：The inverse of a 2x2 matrix \(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) is given by \(\frac{1}{ad-bc} \left[\begin{array}{ll} d & -b \\ -c & a \end{array}\right]\), provided that \(ad-bc\) is not equal to zero. If \(ad-bc\) equals to zero, the matrix does not have an inverse.

Step 3 ：In this case, \(a=8\), \(b=3\), \(c=5\), and \(d=2\). We need to calculate \(ad-bc\) and check if it's not zero.

Step 4 ：Calculating, we find that \(ad-bc = (8*2) - (3*5) = 16 - 15 = 1\). Since this is not zero, the matrix does have an inverse.

Step 5 ：We can now calculate the inverse using the formula above: \(A^{-1} = \frac{1}{1} \left[\begin{array}{ll} 2 & -3 \\ -5 & 8 \end{array}\right] = \left[\begin{array}{ll} 2 & -3 \\ -5 & 8 \end{array}\right]\)

Step 6 ：\(\boxed{\text{Final Answer: The inverse of the matrix } \left[\begin{array}{ll} 8 & 3 \\ 5 & 2 \end{array}\right] \text{ is } \left[\begin{array}{ll} 2 & -3 \\ -5 & 8 \end{array}\right]}\)