<s>[INST] Existence and uniqueness of $A\in M_2(\mathbb R)$ such that $(A^{-1} - 3 I_2)^t = 2 \begin{pmatrix} -1 & -2 \\ 1 & -5 \end{pmatrix}$ I recently thought about this exercise that states:
Does there exist a matrix $A \in M_2(\mathbb R)$ s.t. 
$$(A^{-1} - 3 I_2)^t = 2 \left( \begin{array}{ccc}
-1 & -2   \\
1 & -5  \\
 \end{array} \right)$$
But such a matrix $A$ is not uniquely defined?
I thought that the determinant of $A$ must have determinant = 3 to have a matrix that is not uniquely defined. But I am a bit lost on how to actually find it.
Anyone mind lending a hand?
 [/INST] $$(A^{-1} - 3 I_2)^t = 2 \left( \begin{array}{ccc}
-1 & -2   \\
1 & -5  \\
 \end{array} \right)$$
$$(A^{-1} - 3 I_2) = 2 \left( \begin{array}{ccc}
-1 & 1   \\
-2 & -5  \\
 \end{array} \right)=\begin{pmatrix} -2 & 2 \\ -4 & -10\end{pmatrix}$$
$$A^{-1} =\begin{pmatrix} 1 & 2 \\ -4 & -7\end{pmatrix}$$
$$A=\begin{pmatrix} 1 & 2 \\ -4 & -7\end{pmatrix}^{-1}$$
Are you able to compute $A$ explicitly?
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