The purpose of this question is to show that $\sqrt { 3 }$ is not an eigenvalue of a matrix in $S _ { 2 } ( \mathbb { Q } )$. We assume that there exists $M \in S _ { 2 } ( \mathbb { Q } )$ such that $\sqrt { 3 }$ is an eigenvalue of $M$. 2a. Using the irrationality of $\sqrt { 3 }$, show that the characteristic polynomial of $M$ is $X ^ { 2 } - 3$. 2b. Show that if $n \in \mathbb { Z }$, then $n ^ { 2 }$ is congruent to 0 or 1 modulo 3. 2c. Show that there does not exist a triple of integers $( x , y , z )$ that are coprime as a whole such that $x ^ { 2 } + y ^ { 2 } = 3 z ^ { 2 }$. 2d. Conclude.
The purpose of this question is to show that $\sqrt { 3 }$ is not an eigenvalue of a matrix in $S _ { 2 } ( \mathbb { Q } )$. We assume that there exists $M \in S _ { 2 } ( \mathbb { Q } )$ such that $\sqrt { 3 }$ is an eigenvalue of $M$.
2a. Using the irrationality of $\sqrt { 3 }$, show that the characteristic polynomial of $M$ is $X ^ { 2 } - 3$.
2b. Show that if $n \in \mathbb { Z }$, then $n ^ { 2 }$ is congruent to 0 or 1 modulo 3.
2c. Show that there does not exist a triple of integers $( x , y , z )$ that are coprime as a whole such that $x ^ { 2 } + y ^ { 2 } = 3 z ^ { 2 }$.
2d. Conclude.