Which of the following matrices are of the form $P^tAP$ for a suitable $2\times 2$ invertible matrix $P$ over $\mathbb Q$?

Question

Let the matrix $A=\begin{pmatrix} 0& 1 \\ 1 & 0\end{pmatrix}$ over the field $\mathbb Q$ of rationals.Which of the following matrices are of the form $P^tAP$ for a suitable $2\times 2$ invertible matrix $P$ over $\mathbb Q$?

1. $\begin{pmatrix} 2& 0 \\ 0 & -2\end{pmatrix}$

2. $\begin{pmatrix} 2& 0 \\ 0 & 2\end{pmatrix}$

3. $\begin{pmatrix} 1& 0 \\ 0 & -1\end{pmatrix}$

4. $\begin{pmatrix} 3& 4 \\ 4 & 5\end{pmatrix}$

My Try:-

Characteristic equation of $A$ is $\lambda^2-1=0.$ So, Distinct Eigen values. So, It is diagonalizable. Hence there is an invertible $P$ such that $P^tAP=\begin{pmatrix} 1& 0 \\ 0 & -1\end{pmatrix}.$ It was a question appeared in CSIR Exam. I get less than 5 minutes to solve this problem. I can take $P=\begin{pmatrix} a& b \\ c & d\end{pmatrix}$ and Solve $P^tAP=\begin{pmatrix} 3& 4 \\ 4 & 5\end{pmatrix}$, $P^tAP=\begin{pmatrix} 2& 0 \\ 0 & 2\end{pmatrix}$ and $P^tAP=\begin{pmatrix} 2& 0 \\ 0 & -2\end{pmatrix}.$ But it is time consuming. Can you suggest some theory behind solving this?

Answer 1

A is a diagonalisable matrix with distinct eigenvalues. So the matrix which can be written as P^tAP have also diagonalisable having distinct eigenvalues. See that only option (2) does not have distinct eigenvalues. Hence answer will be rest others.