Let $R$ be a commutative ring with $1$.
Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$.
Is it true that $A$ is also a Noetherian ring?

A square matrix $A$ is called idempotent if $A^2=A$.

(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.

Prove that $P$ is an idempotent matrix.

(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.

Prove that $Q$ is an idempotent matrix.

(c) Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).

Let $A$ be a square matrix such that
\[A^{\trans}A=A,\]
where $A^{\trans}$ is the transpose matrix of $A$.
Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.

A square matrix $A$ is called idempotent if $A^2=A$.

(a) Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix.

(b) Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then determine all integers $k$ such that the matrix $I-kA$ is idempotent.

(c) Let $A$ and $B$ be $n\times n$ matrices satisfying
\[AB=A \text{ and } BA=B.\]
Then prove that $A$ is an idempotent matrix.

Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$.
We assume that $A$ is idempotent, that is, $A^2=A$.
Then prove that
\[\R^n=\im(T) \oplus \ker(T).\]

For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions.

$A=aP+(a+1)Q$

$P^2=P$

$Q^2=Q$

$PQ=O$

$QP=O$,

where $O$ is the $2\times 2$ zero matrix.
Then do the following problems.

(a) Prove that $(P+Q)A=A$.

(b) Suppose $a$ is a positive real number and let
\[ A=\begin{bmatrix}
a & 0\\
1& a+1
\end{bmatrix}.\]
Then find all matrices $P, Q$ satisfying conditions (1)-(5).

(c) Let $n$ be an integer greater than $1$. For any integer $k$, $2\leq k \leq n$, we define the matrix
\[A_k=\begin{bmatrix}
k & 0\\
1& k+1
\end{bmatrix}.\]
Then calculate and simplify the matrix product
\[A_nA_{n-1}A_{n-2}\cdots A_2.\]

Suppose the following information is known about a $3\times 3$ matrix $A$.
\[A\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}=6\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix},
\quad
A\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}, \quad
A\begin{bmatrix}
2 \\
-1 \\
0
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}.\]

(a) Find the eigenvalues of $A$.

(b) Find the corresponding eigenspaces.

(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?