The crash happened just after 4:30 p.m. in the 1300 block of West 46 Street in the Vermont Square neighborhood.
When firefighters arrived to the scene, they found a person trapped between the SUV and the wall of the building, according to the L.A. Fire Department.
The victim died at the scene. Her husband was close by and could be seen clutching on to her body.
The driver stayed at the scene and was later arrested after police said evidence pointed to him driving recklessly at the time of the crash.
Safety inspectors were called in to check if the building was still safe to occupy.
City News Service contributed to this report.