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[toanA37]

P3.
Let A be an $nxn$ matrix with integer entries and$b_{1}, b_{2}...b_{k}$ be integers satisfying $detA =b_{1}b_{2}...b_{k}$. Prove that there exist $n x n$ matrices $B_{i}$ with integers entries such that $A=B_{1}B_{2}...B_{k}$ and$detB_{i}=b_{i}$ for all $ i =1,2,...,k $ .
P12
Let $A_{i},B_{i},S_{i} (i= 1, 2, 3)$ be invertible real $2 x 2$ matrices such that

* not all $A_{i}$ have a common real eigenvector,
* $A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i= 1, 2, 3$,
* $A_{1}A_{2}A_{3}=B_{2}B_{2}B_{3}=I$

Prove that there is an invertible $2 x 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1, 2, 3$ .