What is an eigenvalue in the equation \(A\mathbf v=\lambda\mathbf v\)?

Why must \(A\) be square when discussing eigenvalues?

Find the eigenvalues of \(D=\begin{pmatrix}2&0\\0&7\end{pmatrix}\).

Write the characteristic equation for \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\).

Find the eigenvalues of \(A=\begin{pmatrix}3&0\\0&-1\end{pmatrix}\) using the characteristic equation.

Find the eigenvalues of \(A=\begin{pmatrix}1&4\\0&5\end{pmatrix}\).

Check whether \(4\) is an eigenvalue of \(A=\begin{pmatrix}4&1\\0&2\end{pmatrix}\).

Check whether \(-1\) is an eigenvalue of \(A=\begin{pmatrix}2&0\\0&3\end{pmatrix}\).

Find the eigenvalues of \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\).

Find the eigenvalues of \(A=\begin{pmatrix}0&1\\-2&3\end{pmatrix}\).

A matrix has characteristic polynomial \(p(\lambda)=(\lambda-2)^2(\lambda+5)\). List the eigenvalues and algebraic multiplicities.

Explain why \(\det A=0\) means \(0\) is an eigenvalue of square matrix \(A\).

For which \(k\) does \(A=\begin{pmatrix}k&0\\0&2\end{pmatrix}\) have repeated eigenvalue \(2\)?

For which \(a\) is \(3\) an eigenvalue of \(A=\begin{pmatrix}a&1\\0&4\end{pmatrix}\)?

Find all \(t\) such that \(A=\begin{pmatrix}1&t\\t&1\end{pmatrix}\) has eigenvalue \(0\).

For which \(c\) does \(A=\begin{pmatrix}2&c\\0&2\end{pmatrix}\) have only one distinct eigenvalue?

A repeated eigenvalue is not always a problem for finding eigenvalues. Explain using \(A=\begin{pmatrix}5&1\\0&5\end{pmatrix}\).

A student solves \(\det A=0\) to find all eigenvalues. Diagnose the mistake.

Prove that eigenvalues of a diagonal matrix are its diagonal entries.

Explain why a real rotation by \(90^\circ\), \(R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\), has no real eigenvalues.