Fibonacci-style matrix expansion For each row and then column of a matrix, we can add an extra entry with the sum of the last two entries in that row or column.  For example with the following input matrix:
[ 1 1 1 ]
[ 2 3 4 ]

The resulting matrix would be:
[ 1 1 1 2 ]
[ 2 3 4 7 ]
[ 3 4 5 9 ]

Given an input of an integer N and an [X,Y] matrix of size at least 2x2, perform the above expansion N times and output the result.  The resulting matrix will always be of size [X+N,Y+N].
Examples:
Input:                     Output:

2, [ 0 0 ]                 [ 0 0 0 0 ]
   [ 0 0 ]                 [ 0 0 0 0 ]
                           [ 0 0 0 0 ]
                           [ 0 0 0 0 ]


3, [ 1 1 1 ]               [ 1  1  1  2  3  5 ]
   [ 2 3 4 ]               [ 2  3  4  7 11 18 ]
                           [ 3  4  5  9 14 23 ]
                           [ 5  7  9 16 25 41 ]
                           [ 8 11 14 25 39 64 ]
MATL, 13 14 15 16 20 21 bytes

2*:"!tP2:Y)sv


Thanks @Zgarb for removing 1 byte!

Try it online!

2*         % implicitly input number N and multiply by 2
:          % create vector [1,2,...,2*N]
"          % for loop: do this 2*N times
  !        %   transpose. Implicitly input matrix in the first iteration
  tP       %   duplicate and flip vertically
  2:       %   vector [1,2]
  Y)       %   pick submatrix formed by the first two rows
  s        %   sum of each column
  v        %   append as a new row
           % end for
           % implicit display