## Idempotent Matrices. 2007 University of Tokyo Entrance Exam Problem

## Problem 265

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

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

(*Tokyo University Entrance Exam 2007*)

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