# Tagged: U.Tokyo.LA

## Problem 265

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

1. $A=aP+(a+1)Q$
2. $P^2=P$
3. $Q^2=Q$
4. $PQ=O$
5. $QP=O$,

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
$A_nA_{n-1}A_{n-2}\cdots A_2.$

(Tokyo University Entrance Exam 2007)

## Problem 43

Let $a$ and $b$ be two distinct positive real numbers. Define matrices
$A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.$

Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a non-zero real matrix satisfying the relation
$AX+XB=\lambda X. \tag{*}$

(The University of Tokyo Linear Algebra Exam)

## Problem 42

Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix} -1 \\ 2 \\ 0 \\ -1 \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$.
Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional.

(a) Find an orthonormal basis for the eigenspace of the eigenvalue $2$ of $A$.

(b) Find $A\mathbf{v}$, where
$\mathbf{v}=\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}.$

(The University of Tokyo Linear Algebra Exam)