The second-year forward was examined via ultrasound on Tuesday by Dr. Luga Podesta of the Kerlan-Jobe Orthopaedic Clinic, revealing the tear that will knock out Kelly for at least six weeks.
Kelly missed most of the preseason and all but three games of the regular season with hamstring issues.
In his brief time on the floor, Kelly has averaged 3.7 points and 1.7 rebounds.
The Lakers have struggled to avoid injuries this season with Steve Nash (back) and Julius Randle (leg) out for the season. Nick Young (thumb) has yet to play but could make his debut on Tuesday.
Additionally, Wayne Ellington is on leave after the recent death of his father.
Xavier Henry, who has been healthy enough to stay on the active roster, is nowhere near the same player he was last season after knee, wrist and back injuries.
The Lakers are 1-9. They'll open a three-game road trip in Atlanta against the Hawks on Tuesday.