Whisk flour, baking powder, baking soda and salt in a medium bowl. Put the butter and sugar into the bowl of an electric mixer fitted with the paddle attachment. Mix on medium-high speed until pale and fluffy, 3 to 4 minutes. Reduce speed to medium. Mix in eggs, one at a time. Mix in flour mixture in 3 batches, alternating with sour cream. Add vanilla, and mix 1 minute.

Portion out 1/3 of the batter into a small bowl. Mix in matcha powder, stirring well to combine.

Spread one-third of the plain batter into prepared pan. Use a small offset spatchula to ensure an even, flat layer. Dollop with 1/3 of matcha batter and use a spatchula to gently spread the matcha layer on the top of the other. Spread another third of the plain batter on top, followed by another third of the matcha batter. Repeat one more time so that the final layer is the rest of the matcha batter. Run a thin knife through batter to marbleize. Run spatchula over top to ensure the batter is flat in the end.

Bake until a cake tester inserted into center comes out clean, 50 to 55 minutes. Let cool in pan on a wire cooking rack for a few minutes. Lift cake out and allow to cool completely on rack.