How do you set every other row as well as every other column to zero?

To remove every other column, set it to empty. You can do "every other" generally using A:2:B where A is the first value and B is the last value, and you can stick this in as an index.

a=reshape(1:25,5,5)
a = 5×5
     1     6    11    16    21
     2     7    12    17    22
     3     8    13    18    23
     4     9    14    19    24
     5    10    15    20    25
a1=a;
a1(1:2:end,:)=[] % remove every other row
a1 = 2×5
     2     7    12    17    22
     4     9    14    19    24
a2=a;
a2(2:2:end,:)=[] % remove every other row starting at the second row
a2 = 3×5
     1     6    11    16    21
     3     8    13    18    23
     5    10    15    20    25
a3=a;
a3(:,1:2:end)=[] % remove every other column
a3 = 5×2
     6    16
     7    17
     8    18
     9    19
    10    20
a4=a;
a4(:,1:2:end)=0 % set every other row/column to 0
a4 = 5×5
     0     6     0    16     0
     0     7     0    17     0
     0     8     0    18     0
     0     9     0    19     0
     0    10     0    20     0
% If you really feel like you have to do both at once, you can. But it
% would be much easier to just do rows and columns sequentially.
a5=a;
[r,c]=meshgrid(1:size(a5,1),1:size(a5,2));
ind=sub2ind(size(a5),r(mod(r,2)==0 | mod(c,2)==0),c(mod(r,2)==0 | mod(c,2)==0));
a5(ind)=0
a5 = 5×5
     1     0    11     0    21
     0     0     0     0     0
     3     0    13     0    23
     0     0     0     0     0
     5     0    15     0    25