# Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable

## Problem 206

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.

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## Proof.

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 $1$.

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