AmMaj9 is A, C, E, G# and B. It keeps every note of and adds the .
AmMaj9 and both contain the 9th. The difference is the 7th: AmMaj9 uses , while uses the flat 7th.
Choose between them by the seventh under a held 9th. The minor 3rd remains in both chords.
Also written AmM9, Am(maj9) or A-Δ9.

AmMaj9 in 1st inversion puts C in the bass instead of A, the 3rd of the chord. The notes are the same; what changes is what the bass line does, which is most of why you would reach for it — it lets the lowest voice move by step where the root would have jumped.
One of AmMaj9’s 9 shapes does this:x-x-10-9-9-7
How you get to it. The turn that lands on AmMaj9 is to : the ii and the V of A minor, which is where a minor tonic is usually arrived at rather than simply begun on.
Hear it land:→→
Contains AmMaj9 and adds eleventh and flat thirteenth.
Contains AmMaj9 and adds eleventh and thirteenth.
The common shapes and the derived voicings come from different places, and both are explained in how this site is built.
Shows each chord the way this style would voice it, and marks which of its alternatives belong to the idiom.