Cover of "The Identity Matrix" by Jack L. Chalker
The cover art is a drawing of a woman below a UFO-style spacecraft.
Back Cover of "The Identity Matrix"
Text: The Invaders are among us
"While backpacking in Alaska, Victor Ganser, a 35-year-old Ph.D. in political science, finds himself trapped in the body of a 13-year-old Tlingit Indian girl. A little later he is forced from there into the spectacularly beautiful body of an 18-year-old blonde. He finds that the switches are part of a skirmish for Earth by two alien races... And that's just the beginning. Like many of Jack Chalker's earlier books, this one is almost totally unpredictable and will keep you guessing to the last page."
I found a copy in a used book store several years ago and picked it up solely to make a Linear Algebra joke about the title.
Anyway, I'm a woman now.