Let A be a symmetric matrix of order 2 with integer entries.

Question

Let $A$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^2$ is 1, then the possible number of such matrices is

Accepted Answer

Let $A=\begin{pmatrix}a&b\b&c\end{pmatrix}$. Then $\operatorname{trace}(A^2)=a^2+2b^2+c^2=1$ (integer entries).

The only integer solutions are: one of $a,c$ is $\pm1$ and the other and $b$ are $0$ --- giving $A=\begin{pmatrix}1&0\0&0\end{pmatrix},\begin{pmatrix}-1&0\0&0\end{pmatrix},\begin{pmatrix}0&0\0&1\end{pmatrix},\begin{pmatrix}0&0\0&-1\end{pmatrix}$.

(Any attempt to use $b=\pm1$ forces $2b^2=2>1$, impossible with integers.)

So there are 4 such matrices.

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