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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.

  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
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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