is this mate possible?

Black king has 5 potential moves and is not in check.

a8 and c8. These squares are covered by pawn the on b7. Black would be putting himself in check.

The other squares are a7,b7, and c7. Again black cannot move to or capture on these squares because again he would be putting himself in check because of whites king. 

The move a7 puts the king in check and since the king cannot capture of move to any of the potential squares it is check mate. The black king can only survive if his Queen or rook can capture the a7 pawn, but they cannot.

ok

In diagram provided, if white to move, then a7#; if black, then Qd4+ leads to mate.  In reality, this position would not occur unless you are an idiot as nobody would leave its Q&R on the sidelines.