Is there a way to sum together lists of data frames within a larger list?

I have a large list (z) containing 3 lists of 10 data frames. I would like to collapse this object into a list of 3 data frames where each data frame is the sum of the 10 prior data frames (think matrix addition). Here is what I am working with, keep in mind that these are fake numbers, as the real data are read in from hundreds of *.csv files

x = rep(1,100)
x = matrix(x,10,10)
x = as.data.frame(x)
y = list(x,x,x,x,x,x,x,x,x,x)
z = list(y,y,y)  

The desired end product would look like this:

x1 = rep(10,100)
x1 = matrix(x,10,10)
y1 = list(x1,x1,x1)

I keep trying stuff along the lines of:

z1 = c()
for (i in 1:3){
  for (j in 1:10){
    z1[[i]] = sum(z[[i]][[j]])
  }
  }

However, this does not yield the desired output. I have also messed around with some of the the apply functions, but to no avail

Thanks in advance for your help!