Since $3$ is an eigenvalue of the matrix $A$, we have
where $I$ is the $2\times 2$ identity matrix.
Thus we have
a-3 & -1\\
a-3 & -1\\
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
2 & -1\\
We compute the characteristic polynomial $p(t)=\det(A-tI)$ of $A$.
2-t & -1\\
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 $A-aI$
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
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 […]
Rotation Matrix in Space and its Determinant and Eigenvalues
For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an […]
Similar Matrices Have the Same Eigenvalues
Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.
We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]