Find the inverse of the matrix $A=\left[\begin{array}{cc}-1 & 1 \\ -2 & -8\end{array}\right]$ or state that there is no inverse.

Step 1 ：Given the matrix $A=\left[\begin{array}{cc}-1 & 1 \\ -2 & -8\end{array}\right]$, we are to find its inverse or state that it does not exist.

Step 2 ：The inverse of a matrix $A$ is given by the formula $A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$, where $\text{det}(A)$ is the determinant of the matrix and $\text{adj}(A)$ is the adjugate of the matrix.

Step 3 ：The adjugate of a 2x2 matrix $A=\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]$ is given by $\text{adj}(A)=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$.

Step 4 ：First, we calculate the determinant of the matrix. If the determinant is zero, then the matrix does not have an inverse. If the determinant is not zero, we can calculate the adjugate of the matrix and then use the formula to find the inverse.

Step 5 ：The determinant of the matrix $A$ is $(-1)\times(-8) - (1)\times(-2) = 8 - (-2) = 10$.

Step 6 ：Since the determinant of the matrix is not zero, the matrix does have an inverse. Now, we can calculate the adjugate of the matrix.

Step 7 ：The adjugate of the matrix $A$ is $\text{adj}(A)=\left[\begin{array}{cc}-8 & -1 \\ 2 & -1\end{array}\right]$.

Step 8 ：Using the formula for the inverse of a matrix, we find that $A^{-1} = \frac{1}{10} \times \left[\begin{array}{cc}-8 & -1 \\ 2 & -1\end{array}\right] = \left[\begin{array}{cc}-0.8 & -0.1 \\ 0.2 & -0.1\end{array}\right]$.

Step 9 ：\(\boxed{\text{Final Answer: The inverse of the matrix } A=\left[\begin{array}{cc}-1 & 1 \\ -2 & -8\end{array}\right] \text{ is } A^{-1}=\left[\begin{array}{cc}-0.8 & -0.1 \\ 0.2 & -0.1\end{array}\right]}.\)