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 $3$?
Since $3$ is an eigenvalue of the matrix $A$, we have
\[0=\det(A-3I),\]
where $I$ is the $2\times 2$ identity matrix.
Thus we have
\begin{align*}
0&=\det(A-3I)\\
&=\begin{vmatrix}
a-3 & -1\\
1& 4-3
\end{vmatrix}\\
&=\begin{vmatrix}
a-3 & -1\\
1& 1
\end{vmatrix}\\
&=(a-3)(1)-(-1)(1)=a-2.
\end{align*}
Thus the value of $a$ must be $2$.
(b) Does the matrix $A$ have eigenvalues other than $3$?
Let us determine all the eigenvalue of the matrix
\[A=\begin{bmatrix}
2 & -1\\
1& 4
\end{bmatrix}.\]
We compute the characteristic polynomial $p(t)=\det(A-tI)$ of $A$.
We have
\begin{align*}
p(t)&=\det(A-tI)\\
&=\begin{vmatrix}
2-t & -1\\
1& 4-t
\end{vmatrix}\\
&=(2-t)(4-t)-(-1)(1)\\
&=t^2-6t+9\\
&=(t-3)^2.
\end{align*}
Since the eigenvalues are roots of the characteristic polynomial, solving $(t-3)^2=0$ we see that $t=3$ is the only eigenvalue of $A$ (with algebraic multiplicity $2$).
Hence the matrix $A$ does not have eigenvalues other than $3$.
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 […]
Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable
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.
Proof.
To find eigenvalues we first compute the characteristic polynomial of the […]
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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 $A$.
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Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.
Proof.
We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]