Preheat the oven to 170 degrees C. Lightly oil a 50cm springform pan.
Beat the eggs and the sugar together until pale and thick.
Sift in the flour, baking powder, salt and bicarbonate of soda. Mix well then add the oil and walnuts. Then the carrot and banana.
Pour the mixture into the springform pan. Bake for 1 1/4 hours or until a skewer inserted into the middle of the cake comes out clean. Cover with foil if the top of the cake begins to brown too quickly. Take out and cool on a rack.
Icing: beat the butter and cream cheese together then add the sugar and vanilla. Spread onto the cold cake. Sprinkle with walnuts.