Question Type 1: Finding the eigenvalues of a 2x2 matrix using the characteristic polynomial Exercises

Determine the eigenvalues of $A=\begin{pmatrix}2 & 1\\1 & 2\end{pmatrix}$ by forming the characteristic equation.

Find the eigenvalues of the matrix $A = \begin{pmatrix}4 & 3\\2 & 1\end{pmatrix}$ using its characteristic polynomial.

Compute the eigenvalues of $A=\begin{pmatrix}0 & 1\\-2 & 3\end{pmatrix}$ by solving its characteristic equation.

Find the eigenvalues of $A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$ using its characteristic polynomial.

Find the eigenvalues of $A=\begin{pmatrix}7 & 4\\6 & 5\end{pmatrix}$ using its characteristic equation.

For the matrix $A=\begin{pmatrix}a & b\\0 & a\end{pmatrix}$ with constants $a,b$, find its eigenvalues. [2 marks]

Determine the eigenvalues of $A=\begin{pmatrix}-1 & 2\\2 & -1\end{pmatrix}$ by finding its characteristic polynomial.

Let $A=\begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}$. Find its eigenvalues in terms of $\theta$.

Find the eigenvalues of the rank-one matrix $A=\begin{pmatrix}1 & 1\\1 & 1\end{pmatrix}$.

Find the eigenvalues of $A=\begin{pmatrix}3 & -4\\2 & -1\end{pmatrix}$ by solving its characteristic polynomial.

Determine the eigenvalues of the rotation matrix $A=\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}$.

Find the eigenvalues of the diagonal matrix $A = \begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix}$.