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I noticed that a positive integer $2\times2$ matrix having unit determinant appears to have only single one largest element. I could not find any counter examples so I would be thankful if someone could explain why is this the case or give a counter example. Consider the following:

$$ \begin{bmatrix} 3 & 2 \\ 1 & 1 \\ \end{bmatrix} $$

$$ \begin{bmatrix} 5 & 2 \\ 2 & 1 \\ \end{bmatrix} $$

$$ \begin{bmatrix} 10 & 3 \\ 3 & 1 \\ \end{bmatrix} $$

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This is true.

Consider the matrix $$ \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ Its determinant is $ad-bc$.

Case 1. $a=b$ (the case $a=c$ is similar)

In this case we have that $ad-ac=1$ or $a(d-c)=1$. For positive integers it is possible only if $a=1, c=1, d=2$. And here is the maximum element.

Case 2. $a=d$

In this case we have that $a^2-bc=1$.

$b$ can't equal $c$, because there are no solutions to $a^2-b^2=1$. Thus, $b>a$ (or $c>a$). Otherwise, there are no solutions to $a^2-bc=1$.

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