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Define a function, either recursive or not, to do that.

Recursive version:

def Mk_rec(M0, k, F):
    import numbers
    if not isinstance(k, numbers.Integral) or k < 0:
        raise ValueError(f'Expected non-negative integer k, got: {k}')
    MS = M0.parent()  # the matrix space
    if k == 0:
        return MS(M0)
    M = Mk_rec(M0, k - 1, F)
    FF = lambda a: F(a, a)
    return MS(lambda i, j: FF(M[i, j]))

Non-recursive version:

def Mk_nrec(M0, k, F):
    import numbers
    if not isinstance(k, numbers.Integral) or k < 0:
        raise ValueError(f'Expected non-negative integer k, got: {k}')
    MS = M0.parent()  # the matrix space
    M = MS(M0)
    FF = lambda a: F(a, a)
    for j in range(k):
        M = MS(lambda i, j: FF(M[i, j]))
    return M

Examples:

sage: M0 = matrix([[0, 2], [1, -1]])
sage: F = lambda a, b: a + b
sage: Mk_nrec(M0, 2, F)
[ 0  8]
[ 4 -4]
sage: Mk_rec(M0, 2, F)
[ 0  8]
[ 4 -4]