The shortest distance from a point to a plane is along the line perpendicular to the plane. The normal vector to x+ y+ z= 0 at any point on the plane is <1, 1, 1>. The line through (1, 0, 0) with direction vector < 1, 1, 1> is x= t+1, y= t, z= t. Where does that line intersect the plane x+ y+ z= 0?

What (a) tells you is that (1, 0, 0) is mapped by A onto (3/2, -1/3, -1/3). Do the same thing to determine what A maps (0, 1, 0) and (0, 0, 1) to. those three vectors form the columns of A in the standard basis.