We show that quantum computers improve on the best known classical algorithms for matrix inversion (and singular value decomposition) as far as space is concerned. This adds to the (still short) list of important problems where quantum computers are of help. Specifically, we show that the inverse of a well conditioned matrix can be approximated in quantum logspace with in- termediate measurements. This should be compared with the best known classical algorithm for the problem that re- quires (log2 n) space. We also show how to approximate the spectrum of a normal matrix, or the singular values of an arbitrary matrix, with ε additive accuracy, and how to approximate the singular value decomposition (SVD) of a matrix whose singular values are well separated. The technique builds on ideas from several previous works, including simulating Hamiltonians in small quantum space (building on  and ), treating a Hermitian matrix as a Hamiltonian and running the quantum phase estimation procedure on it (building on ) and making small space probabilistic (and quantum) computation consistent through the use of offline randomness and the shift and truncate method (building on ).