The equation provided in the image is:

$(A^{-1} - 2I)^T = -2 \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$

Let’s break down the components of this equation:

1. $A^{-1}$: This represents the inverse of matrix $A$.
2. $I$: This represents the identity matrix (which matches the dimensions of matrix $A$).
3. $T$: This indicates the transpose of the matrix.
4. On the right-hand side, we have a scalar multiple of a matrix.

To find $A$, we would have to solve for $A^{-1}$ by equating both sides, which requires taking the transpose and inverse operations.

Steps to approach:

1. Compute the transpose of the right-hand side matrix.
2. Set $A^{-1} - 2I$ equal to the transposed matrix.
3. Solve for $A^{-1}$ and then find $A$ by taking the inverse.

Do you want to proceed with solving this equation? If so, I can compute the solution step-by-step.

Here are five related questions you might explore:

1. What properties of matrix transposition and inversion will apply in this problem?
2. How can we compute the inverse of a 2x2 matrix?
3. What happens when you add or subtract the identity matrix to/from another matrix?
4. What are the conditions for a matrix to have an inverse?
5. How would the solution change if the matrix on the right-hand side was not symmetric?

Tip: When dealing with matrix transposition and inversion, remember that $(A^{-1})^T = (A^T)^{-1}$, which can simplify certain matrix operations.