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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