Security analysis of a "one-time pad" type hill cipher
Expand the equation system corresponding to the matrix multiplication:

$c_j = \sum_{i=0}^{n-1}k_{i,j}p_i$

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.