Ballistic Matrix. It says to place it on the deployment card. Does this mean that it stays with BT through the game?

It says to exhaust it before declaring an attack, but not when to discard. So does it become a readied card at end of round?

All exhausted cards (except class and item cards of heroes in the campaign) ready during the ready step of the status phase.

Ballistic Matrix has an exhaust cost so that you can use it only for one of the attacks of Missile Salvo.

So because it says to exhaust it, that means it does become readied, correct?

I know it would only be for one attack of the salvo, or his primary attack, but just wanted to make sure it didn't get discarded.

Unlike other command cards which are discarded when played, Ballistic Matrix explicitly says to place it on the deployment card. The other ability has exhaust as a cost.

(Discard as a cost would be specified separately like in supply and boon cards. And in skirmish deplete would be equal to discard.)