Security analysis of a “one-time pad” type hill cipher

Suppose the Hill cipher were modified to something like a one-time pad cipher, where Alice wants to send a message to Bob, and she chooses a key matrix randomly everytime a new message is sent (and the inverse of the key matrix is sent to Bob by other means everytime - just as in the one-time pad). Would this yield perfect secrecy or at least $\mathsf{IND-CCA}$?

Any thoughts on how to proceed with the security analysis of this?

-

$c_j = \sum_{i=0}^{n-1}k_{i,j}p_i$
In other words, each element of the cipher text corresponds to the sum of the cipher text elements of an OTP encryption of the input plain text. If the matrix is never reused, it should be fairly easy to go from here. You are basically using $n$ one time pads to encrypt the message instead of a single one. - Almost, since the $k_{i,j}$ values of the matrix have to be selected in such way that the determinant is not zero (presuming a prime field is used for the arithmetic operations). The next step is therefore to observe what this constraint means: If the diagonal is selected at random, the other matrix elements might be selected in such way that the determinant is non-zero.
If the $p_i$ are all zeroes, what are the $c_j$? Is that compatible with OTP-level security? –  fgrieu Mar 4 '12 at 11:51
Now, assuming we work $\bmod{27}$ and $p_j\ne 0$, what is $c_j\bmod 3$ if all $p_i\equiv 0 \pmod 3$? Again, is that compatible with OTP-like security? –  fgrieu Mar 4 '12 at 12:44