This thesis presents a novel algorithm for Transposing Rectangular matrices Inplace and in Parallel (TRIP) including a proof of correctness and an analysis of work, span and parallelism. After almost 60 years since its introduction, the problem of inplace rectangular matrix transposition still does not have a satisfying solution. Increased concurrency in todays computers, and the need for lowoverhead algorithms to solve memoryintense challenges are motivating the development of algorithms like TRIP. The algorithm is based on recursive splitting of the matrix into submatrices, independent, parallel transposition of these submatrices, and subsequent combining of the results by a parallel, perfect shuffle. We prove correctness of the algorithm for different matrix shapes (ratios of dimensions), and analyze work and span: For an $M\times N$ matrix, where $M>N$, and both $M$ and $N$ are powers of two, TRIP has work \[W_1 \trip(M,N)=\Theta\left(MN\log\frac
