Determine all eigenvalues and their algebraic multiplicities of the matrix
\[A=\begin{bmatrix}
1 & a & 1 \\
a &1 &a \\
1 & a & 1
\end{bmatrix},\]
where $a$ is a real number.

To find eigenvalues we first compute the characteristic polynomial of the matrix $A$ as follows.
\begin{align*}
\det(A-tI)&=\begin{vmatrix}
1-t & a & 1 \\
a &1-t &a \\
1 & a & 1-t
\end{vmatrix}\\
&=(1-t)\begin{vmatrix}
1-t & a\\
a& 1-t
\end{vmatrix}-a\begin{vmatrix}
a & a\\
1& 1-t
\end{vmatrix}+\begin{vmatrix}
a & 1-t\\
1& a
\end{vmatrix}
\end{align*}
We used the first row cofactor expansion in the second equality.

After we compute three $2 \times 2$ determinants and simply, we obtain
\[\det(A-tI)=-t(t^2-3t+2-2a^2).\]
The eigenvalues of $A$ are roots of this characteristic polynomial. Thus, eigenvalues are
\[0, \quad \frac{3\pm \sqrt{1+8a^2}}{2}\]
by the quadratic formula.

Now the only possible way to obtain a multiplicity $2$ eigenvalue is when
\[\frac{3- \sqrt{1+8a^2}}{2}=0\]
and it is straightforward to check that this happens if and only if $a=1$.

Therefore, when $a=1$ eigenvalues of $A$ are $0$ with algebraic multiplicity $2$ and $3$ with algebraic multiplicity $1$.

When $a \neq 1$, eigenvalues are
\[0, \quad \frac{3\pm \sqrt{1+8a^2}}{2}\]
and each of them has algebraic multiplicity $1$.

Maximize the Dimension of the Null Space of $A-aI$
Let
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
\end{bmatrix}.\]
Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
Your score of this problem is equal to that […]

Determine a Matrix From Its Eigenvalue
Let
\[A=\begin{bmatrix}
a & -1\\
1& 4
\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is some real number.
Suppose that the matrix $A$ has an eigenvalue $3$.
(a) Determine the value of $a$.
(b) Does the matrix $A$ have eigenvalues other than […]

Find All the Eigenvalues of Power of Matrix and Inverse Matrix
Let
\[A=\begin{bmatrix}
3 & -12 & 4 \\
-1 &0 &-2 \\
-1 & 5 & -1
\end{bmatrix}.\]
Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.
Proof.
We first determine all the eigenvalues of the matrix […]

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For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an […]

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Find the inverse matrix of the $3\times 3$ matrix
\[A=\begin{bmatrix}
7 & 2 & -2 \\
-6 &-1 &2 \\
6 & 2 & -1
\end{bmatrix}\]
using the Cayley-Hamilton theorem.
Solution.
To apply the Cayley-Hamilton theorem, we first determine the characteristic […]

A Matrix Having One Positive Eigenvalue and One Negative Eigenvalue
Prove that the matrix
\[A=\begin{bmatrix}
1 & 1.00001 & 1 \\
1.00001 &1 &1.00001 \\
1 & 1.00001 & 1
\end{bmatrix}\]
has one positive eigenvalue and one negative eigenvalue.
(University of California, Berkeley Qualifying Exam Problem)
Solution.
Let us put […]

Find the Inverse Matrix Using the Cayley-Hamilton Theorem Find the inverse matrix of the matrix
\[A=\begin{bmatrix}
1 & 1 & 2 \\
9 &2 &0 \\
5 & 0 & 3
\end{bmatrix}\]
using the Cayley–Hamilton theorem.
Solution.
To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial $p(t)$ of […]