# Largest element of matrix with unit determinant

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:

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

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

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

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 $$a^2-bc=1$$.