Hill Cipher known plaintext attack I know a plaintext - ciphertext couple of length 6 for a hill cipher where its key is a [3x3] matrix.

Based on what I've read and learned, to attack and crack keys of [n x n], if we know a plaintext - ciphertext duo of length $n^2$ then we have our set of n equations with n variables, and this is generally solvable. However in my case there is only length of $6$ instead of $n^2 = 9$ . 

My question is, how will I solve this problem and find the key? Or, since this is an obvious homework question, what should be my way of thinking? I cannot think of anything else but matrix multiplications and inverses but they do not help me at all.
Well, I went and solved the puzzle using brute force and Maple.  I won't spoil the actual answer, but here are some tips that ought to make the process a bit more quicker.


Solving the linear system modulo 2 gives you the parity of the second and third letters of the unknown plaintext.  Note that all vowels in the English alphabet map to even numbers using the standard Hill cipher encoding!
Conversely, solving the system modulo 13 tells you the fourth letter of the unknown plaintext (up to rot13).  This, together with the mod 2 solution, should help you guess what the second and third letters could be.
Even after correctly guessing the second and third letters, and ruling out any non-invertible decoding matrices, the modulo 2 solution will still not be fully determined.  Thus, some brute force and/or guesswork will be required to decode the remaining letters.  Hopefully, however, one of the possible plaintexts will show up as obviously more plausible than the others.


Ps. One more hint: 007.