Let $A, B, C$ are $2\times 2$ diagonalizable matrices.
The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.
From this information, determine the rank of the matrices $A, B,$ and $C$.
Graphs of characteristic polynomials
Add to solve later Let
\[A=\begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic polynomial:
\[p_A(\lambda)=p_B(\lambda)=-(\lambda-1)(\lambda+2)^2.\]
(a) Find all eigenvectors of $A$.
(b) Find all eigenvectors of $B$.
(c) Which matrix $A$ or $B$ is diagonalizable?
(d) Diagonalize the matrix stated in (c), i.e., find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$ or $B=PDP^{-1}$.
(Stanford University Linear Algebra Final Exam Problem)
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Add to solve later Find the inverse matrix of the matrix
\[A=\begin{bmatrix}
\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt]
\frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt]
-\frac{3}{7} & \frac{6}{7} & -\frac{2}{7}
\end{bmatrix}.\]
Add to solve later Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Add to solve later In this post, we explain how to diagonalize a matrix if it is diagonalizable.
As an example, we solve the following problem.
Diagonalize the matrix
\[A=\begin{bmatrix}
4 & -3 & -3 \\
3 &-2 &-3 \\
-1 & 1 & 2
\end{bmatrix}\]
by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
(Update 10/15/2017. A new example problem was added.)
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Add to solve later Let $A$ be an $n\times n$ matrix with real number entries.
Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.
Add to solve later 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.
Add to solve later Show that eigenvalues of a Hermitian matrix $A$ are real numbers.
(The Ohio State University Linear Algebra Exam Problem)
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Add to solve later Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$.
Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.
Add to solve later 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 dimension times five.
(The Ohio State University Linear Algebra Practice Problem)
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Add to solve later Find the value(s) of $h$ for which the following set of vectors
\[\left \{ \mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
h \\
1 \\
-h
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
1 \\
2h \\
3h+1
\end{bmatrix}\right\}\]
is linearly independent.
(Boston College, Linear Algebra Midterm Exam Sample Problem)
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Add to solve later Let $A$ be a $3 \times 3$ matrix.
Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have
\[A\mathbf{x}=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, A\mathbf{y}=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, A\mathbf{z}=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}.\]
Then find the value of the determinant of the matrix $A$.
Add to solve later Let
\[A=\begin{bmatrix}
1 & -1\\
2& 3
\end{bmatrix}.\]
Find the eigenvalues and the eigenvectors of the matrix
\[B=A^4-3A^3+3A^2-2A+8E.\]
(Nagoya University Linear Algebra Exam Problem)
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Add to solve later 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)
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Add to solve later Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces
\[E_2=\Span\left \{\quad \begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix}
1 \\
2 \\
1 \\
1
\end{bmatrix},\quad \begin{bmatrix}
1 \\
1 \\
1 \\
2
\end{bmatrix} \quad\right\}.\]
Calculate $C^4 \mathbf{u}$ for $\mathbf{u}=\begin{bmatrix}
6 \\
8 \\
6 \\
9
\end{bmatrix}$ if possible. Explain why if it is not possible!
(The Ohio State University Linear Algebra Exam Problem)
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Add to solve later Let $A$ be an $n\times n$ matrix. Suppose that $\lambda_1, \lambda_2$ are distinct eigenvalues of the matrix $A$ and let $\mathbf{v}_1, \mathbf{v}_2$ be eigenvectors corresponding to $\lambda_1, \lambda_2$, respectively.
Show that the vectors $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent.
Add to solve later Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$.
Is it true that
\[\det(A+B)=\det(A)+\det(B)?\]
If so, then give a proof. If not, then give a counterexample.
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Add to solve later Let $A=(a_{ij})$ be an $n \times n$ matrix.
We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have
\[a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1\]
for $1 \leq i, j \leq n$.
Let $A=(a_{ij})$ be an $n\times n$ right stochastic matrix. Then show the following statements.
(a)The stochastic matrix $A$ has an eigenvalue $1$.
(b) The absolute value of any eigenvalue of the stochastic matrix $A$ is less than or equal to $1$.
Add to solve later Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix.
If
\[P^{-1}AP=\begin{bmatrix}
1 & 2 & 3 \\
0 &4 &5 \\
0 & 0 & 6
\end{bmatrix},\]
then find all the eigenvalues of the matrix $A^2$.
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Add to solve later Let $A$ be an $n \times n$ matrix. Suppose that the matrix $A^2$ has a real eigenvalue $\lambda>0$. Then show that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.
Add to solve later 
