IB Mathematics Applications & Interpretation Question Type 1: Finding the Eigenvalues of a 2x2 Matrix Using the Characteristic Polynomial Exercises

Determine the eigenvalues of $A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$ by forming the characteristic equation.

Find the eigenvalues of the matrix $A=\left(\begin{array}{cc}4& 3\\ 2& 1\end{array}\right)$ using its characteristic polynomial.

Compute the eigenvalues of $A=\left(\begin{array}{cc}0& 1\\ -2& 3\end{array}\right)$ by solving its characteristic equation.

Find the eigenvalues of $A=\left(\begin{array}{cc}4& 7\\ 2& 6\end{array}\right)$ using its characteristic polynomial.

Find the eigenvalues of $A=\left(\begin{array}{cc}7& 4\\ 6& 5\end{array}\right)$ using its characteristic equation.

For the matrix $A=\left(\begin{array}{cc}a& b\\ 0& a\end{array}\right)$ with constants $a,b$, find its eigenvalues. [2 marks]

Determine the eigenvalues of $A=\left(\begin{array}{cc}-1& 2\\ 2& -1\end{array}\right)$ by finding its characteristic polynomial.

Let $A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$. Find its eigenvalues in terms of $\theta$.

Find the eigenvalues of the rank-one matrix $A=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$.

Find the eigenvalues of $A=\left(\begin{array}{cc}3& -4\\ 2& -1\end{array}\right)$ by solving its characteristic polynomial.

Determine the eigenvalues of the rotation matrix $A=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$.

Find the eigenvalues of the diagonal matrix $A=\left(\begin{array}{cc}5& 0\\ 0& 3\end{array}\right)$.