Appending vectors in a forloop dependent on iteration.

For a m × m matrix A and a fixed vector v ∈ Rn, I am asked to compute the time it takes to compute Av in Matlab, and plot such time for m = 2j, j = 1, 2, . . . , 12. So pretty much the size of the matrix increases in multiples of 2 and as a result I would have to change the size of the vector to make sure dimensions match. Here is what I have so far, however I do not know the syntax for adding 2^(i-1) entries to the vector v for each iteration. (I did the math and this should allow the matrices to multiply without an error. Here is the code I have so far:

tl;dr I need to append 2^(i-1) entries to my vector v after each iteration. I also am not sure if the way I am plotting it makes sense.

v = [1, 2]'                         % I fix a 2x1 vector. 
A = randint(2,2,[0,100])            % I generate a random 2x2 matrix.
time = zeros(1,12);                 % I create a 1x12 vector. 

for i=1:12  
    if i==1                         % When i=1, I could just multiply the 
        tic;                        % 2x2 matrix and the 2x1 vector. 
        w = A*v; 
        t1 = toc; 
        time = [time toc]           % Add the cpu time for i=1.  
    else 
        tic; 
        A = randint(2^(i), 2^(i), [0,100]); %This will allow me to generate 
                                            %4x4 matrices, 8x8 matrices
                                            %etc. 
        %How do I add 2^(i-1) entries to v that are random? I need to do 
        %this in order to make the dimensions match when I multiply the 
        %vector and the matrix together. 
        w = A*v; 
        time2 = toc; 
        time = [time time2]         % Add the cpu time for i=2....12. 

      end

plot([1:12], time)                  % Let the x-variable be the value of i 
                                    %and let the cpu time be on the y-axis.

The mathworks community is great and any and all help will be greatly appreciated. Sincerely, -Andrew.