minimise a linear function of \mathbf{x} where \mathbf{x} is a K\times 1 column vector (all components greater than or equal to zero) and also det(\mathbf{I}+x_1\mathbf{A}_1+\cdots x_K\mathbf{A}_K)> a certain floor, where \mathbf{A}_i are M\times M are given positive definite square matrices.

Can you give some rough idea about the complexity of the problem with respect both K and M? Is it square or cube with respect to them?

If you want a theoretical answer, then you just convert your problem to standard SDP i.e.

Next you look up the complexity of solving such an SDP. I guess that should be in the book of Boyd and Vanderberhe.
My guess is that the worst case complexity will be cubic in M. I mean checking positive definiteness will be cubic in M.

If you want to know what the practical running time is then answer would be along the line

It can indeed be written as a standard SDP. Though as you have written it, it is not even convex. You need to take the $n$th root of the determinant to make it so.