Let $A$ be a $2 \times 2$ matrix such that $\text{trace}(A) = \det(A) = a (a \neq 0)$. Then the trace of $A^{-1}$ is}
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For any $2 \times 2$ matrix:
$\text{trace}(A^{-1}) = \frac{\text{trace}(A)}{\det(A)}$.
Since we are given $\text{trace}(A) = \det(A)$, their ratio is always 1, regardless of the value of $a$ (as long as $a \neq 0$).