A320 Spoiler 3 at status page
 

why after engines shut down, system always shows SPLR 3 as Inop (amber) at status page ?

Spoiler 3 on the A320 is actuated using the blue system.
When you shut the engines down the blue system electric pump shuts off straight away so you loose hydraulic power to spoiler 3 very quickly. The system senses this and displays as an inoperable item on the status page (spoiler 3 ECAM is suppressed).

Because the green and yellow systems are fed off of the engine driven hydraulic pumps they take longer to loose power. These systems will quite happily work at 3000 psi with 10% N2 (pretty much windmilling speed). So while they are running down the hydraulic surfaces could still be moved and so are sensed as operative.
The system reconfigures flight phases to "engines shut down" and the spoiler 3 inoperable disappears.

Lots of system reconfiguration at engine shutdown. This is the same reason why sometimes you get the ENG 1(2) FAIL ECAM as you shut the engines down temporarily. This was/is due to an overlap between flight phases 9/10 so temporarily the engine is sensed as failed before entering phase 10 where this ECAM is suppressed.

Thank you so much it was so clear