Let $M$ be an $2 \times 2$ matrix, with all entries in $\mathbb{N}$: $$ M= \begin{bmatrix} a & b \\ c & d \end{bmatrix} \;. $$ So $$ \mathrm{det}(M) = a d - b c \; . $$ The Euclidean norm (a.k.a. Frobenius Norm) of $M$ is the square root of the sum of the squares of its entries: $$ |M| = \sqrt{ a^2 + b^2 +c^2 + d^2 } \;. $$ I am trying to understand when $\mathrm{det}(M) = |M| = r$, $r \in \mathbb{N}$. For example, when $$ M= \begin{bmatrix} 15 & 9 \\ 3 & 3 \end{bmatrix} \;, $$ we have $$\mathrm{det}(M) = 15 \cdot 3 - 9 \cdot 3 = 18$$ and $$|M| = \sqrt{225 + 81 + 9 + 9} = \sqrt{324} = 18 \;.$$ Another solution is $$ M= \begin{bmatrix} 5 & 11 \\ 49 & 137 \end{bmatrix}\;, \;\mathrm{det}(M) = |M| = 146 \;. $$
Q. What are the solutions for $M$ a $2 \times 2$ matrix and $\mathrm{det}(M) = |M|$ a natural number $r$?
The positive-orthant point $(a,b,c,d)$ lies on the origin-centered sphere of radius $r$ in $\mathbb{R}^4$, but with the added constraint that $a d - b c = r$.
My goal was to understand when $\mathrm{det}(M) = |M|$ is a natural number $r$ for $M$ an $n \times n$ matrix of entries in $\mathbb{N}$, but already for $n=2$ it seems not entirely straightforward.