Perform the operation $A^{2}$ with the given matrices.
\[
A=\left[\begin{array}{cc}
-6 & 3 \\
-1 & -8
\end{array}\right]
\]

Step 1 ：We are given the matrix \(A = \left[\begin{array}{cc} -6 & 3 \\ -1 & -8 \end{array}\right]\)

Step 2 ：We are asked to perform the operation \(A^{2}\), which means multiplying the matrix \(A\) by itself.

Step 3 ：In order to multiply two matrices, we use the formula for matrix multiplication, which is: \((A \cdot B)_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}\), where \(A\) and \(B\) are matrices, \(a_{ik}\) is the element in the \(i\)th row and \(k\)th column of \(A\), and \(b_{kj}\) is the element in the \(k\)th row and \(j\)th column of \(B\).

Step 4 ：In this case, since we are multiplying \(A\) by itself, \(a_{ik}\) and \(b_{kj}\) are the same.

Step 5 ：Performing the multiplication, we find that \(A^{2} = \left[\begin{array}{cc} 33 & -42 \\ 14 & 61 \end{array}\right]\)

Step 6 ：Final Answer: The result of the operation \(A^{2}\) is the matrix \(\boxed{\left[\begin{array}{cc} 33 & -42 \\ 14 & 61 \end{array}\right]}\)