# a question of matrix equations

#### rulin

Prove:

1. Matrix equation $$\displaystyle AXA=A$$ has a solution always for every matrix $$\displaystyle A$$.

2. If both matrix equations $$\displaystyle AY=C$$ and $$\displaystyle ZB=C$$ have solution, then equation $$\displaystyle AXB=C$$ has a solution.

#### dwsmith

MHF Hall of Honor
1. Matrix equation $$\displaystyle AXA=A$$ has a solution always for every matrix $$\displaystyle A$$.
I think this would only work if A is nonsingular and X is the inverse.

#### dwsmith

MHF Hall of Honor
Prove:
2. If both matrix equations $$\displaystyle AY=C$$ and $$\displaystyle ZB=C$$ have solution, then equation $$\displaystyle AXB=C$$ has a solution.
I don't get it. The multiplication is defined by what you wrote.