Help with log det expression

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Have you proven that your optimization problem is convex? Why isn't CVX accepting my model? READ THIS FIRST!

The argument of log_det is concave, but not affine, which it must be in order to be accepted by CVX.

help log_det

log_det Logarithm of the determinant of an SDP matrix.
For a square matrix X, log_det(X) returns
LOG(DET(X))
if X is symmetric (real) or Hermitian (complex) and positive semidefinite,
and -Inf otherwise.

When used in a CVX model, log_det(X) causes CVX's successive 
approximation method to be invoked, producing results exact to within
the tolerance of the solver. Therefore, whenever possible, the use of
DET_ROOTN(X) is to be preferred, because the latter choice can be
represented exactly in an SDP. For example, the objective
    MAXIMIZE(log_det(X))
can be (and should be) replaced with
    MAXIMIZE(DET_ROOTN(X))
in fact, log_det(X) is implemented simply as N*LOG(DET_ROOTN(X)).

Disciplined convex programming information:
    log_det is concave and nonmonotonic; therefore, when used in
    CVX specifications, its argument must be affine.