alkali metal or alkaline earth metal?

No, it must NOT be an alkaline earth metal.

Although some alkaline earth metals (such as Ca, Sr, Ba) can react with water to give hydrogen gas, all alkaline earth metals are not soft metals. This is because the metallic bonds in alkaline earth metals are much stronger than those of alkali metals.

Since element A is a soft metal, the element must NOT be an alkaline earth metal.

Yes.

It can be calcium which is an alkaline earth metal. Calcium is also a soft silvery solid at room conditions. See

http://en.wikipedia.org/wiki/Calcium

It can react with water to give off hydrogen and form an alkaline solution calcium hydroxide.

Ca + 2H2O ---> Ca( OH )2 + H2

Hope it helps.