# Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials

## Problem 609

Let $A$ be a $2\times 2$ real symmetric matrix.
Prove that all the eigenvalues of $A$ are real numbers by considering the characteristic polynomial of $A$.

## Proof.

Let $A=\begin{bmatrix} a& b \\ c& d \end{bmatrix}$.
Then as $A$ is a symmetric matrix, we have $A^{\trans}=A$.
This implies that
$\begin{bmatrix} a& c \\ b& d \end{bmatrix}=\begin{bmatrix} a& b \\ c& d \end{bmatrix}.$ Hence we have $b=c$ by comparing entries.

Now, we find the characteristic polynomial $p(t)$ of $A$.
We have
\begin{align*}
p(t)&=\det(A-t I)=\begin{vmatrix}
a-t & b\\
b& d-t

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