Let $\lambda$ be an arbitrary eigenvalue of $A$.
Then the matrix $A-\lambda I$ is singular, where $I$ is the $2\times 2$ identity matrix.
This is equivalent to having $\det(A-\lambda I)=0$.

We compute the determinant as follows.
We have
\begin{align*}
\det(A-\lambda I)&=\begin{vmatrix}
a-\lambda & b\\
-b& a-\lambda
\end{vmatrix}\\[6pt]
&=(a-\lambda)^2-b(-b)\\
&=a^2-2a\lambda+\lambda^2+b^2\\
&=\lambda^2-2a\lambda+a^2+b^2.
\end{align*}

We solve the equation $\lambda^2-2a\lambda+a^2+b^2=0$ by the quadratic formula and obtain
\begin{align*}
\lambda &=\frac{2a\pm\sqrt{4a^2-4(a^2+b^2)}}{2}=\frac{2a\pm\sqrt{-4b^2}}{2}\\[6pt]
&=a\pm |b|i.
\end{align*}

Since $b\neq 0$ by assumption, the eigenvalue $\lambda=a\pm|b|i$ is not a real number.
As $\lambda$ is an arbitrary eigenvalue of $A$, we conclude that all eigenvalues of $A$ are not real numbers.

Common Eigenvector of Two Matrices and Determinant of Commutator
Let $A$ and $B$ be $n\times n$ matrices.
Suppose that these matrices have a common eigenvector $\mathbf{x}$.
Show that $\det(AB-BA)=0$.
Steps.
Write down eigenequations of $A$ and $B$ with the eigenvector $\mathbf{x}$.
Show that AB-BA is singular.
A matrix is […]

Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues
Let $A$ be an $n\times n$ real matrix.
Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.
We give two proofs.
Proof 1.
Let $\mathbf{x}$ be an eigenvector corresponding to the […]

Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$
Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.
(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.
(b) Let
\[A^{100}=aA^2+bA+cI,\]
where $I$ is the $3\times 3$ identity matrix.
Using the […]

Eigenvalues of Similarity Transformations
Let $A$ be an $n\times n$ complex matrix.
Let $S$ be an invertible matrix.
(a) If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.
(b) If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an […]

There is at Least One Real Eigenvalue of an Odd Real Matrix
Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.
Prove that the matrix $A$ has at least one real eigenvalue.
We give two proofs.
Proof 1.
Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$.
It is a degree $n$ […]

Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix
Consider the matrix $A=\begin{bmatrix}
a & -b\\
b& a
\end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$.
(a) Find all eigenvalues of $A$.
(b) For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$.
(c) Diagonalize the matrix $A$ by finding a […]

A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix
Prove that the matrix
\[A=\begin{bmatrix}
0 & 1\\
-1& 0
\end{bmatrix}\]
is diagonalizable.
Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix.
That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]

Find All Values of $x$ so that a Matrix is Singular
Let
\[A=\begin{bmatrix}
1 & -x & 0 & 0 \\
0 &1 & -x & 0 \\
0 & 0 & 1 & -x \\
0 & 1 & 0 & -1
\end{bmatrix}\]
be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.
Hint.
Use the fact that a matrix is singular if and only […]