The Time Trapper created a pocket universe with its own Earth as part of a plan to manipulate the Legion of Super-Heroes. After Superboy went onto the future to save the world, Lex Luthor, a noble scientist, created a lifeform from protomatter with Lana Lang's genetic material to aid them in the war against three rogue Kryptonians, General Zod, Zaora and Quex-Ul.
The matrix's first appearance was Superman 21. This Superman title spun out of the Crisis on Infinite Earths series.
Solo Adventures / Teen Titans
The shape changing Matrix was dispatched to a parallel world, hoping Superman would help. He answered the call and was forced to kill the three Kryptonians to save their world. Matrix was reduced to an amorphous mass in the fight. Superman brought Matrix to his world to recover on the Kent Farm. Matrix took on a feminine form and paid homage to her savior by using her powers to become Supergirl. Due to her naivety, she fell under Lex Luthor's influence for a while, until she saw the manipulative criminal he is. She also had a brief stint in the Teen Titans.
Matrix saved the life of Linda Danvers, by merging their life essences. A new being was created with new abilities such as increased strength but reduced shape changing.
In time she discovered the violent life Linda had led and sought to redeem her. During her encounters with various supernatural entities, she discovered she was a reborn Earth-Angel. One of three on Earth.
She also lost her shape-shifting abilities as she began to see herself as just Linda or Supergirl. The part of her protoplasmic form that didn't merge became an insane creature and fought Linda/Matrix until she was absorbed by the Angel of Light, Blithe.
Matrix and Linda were eventually separated by Demon Queen Lilith. Matrix was guarded by a woman called Twilight. When Linda rescued Matrix after fighting and then gaining Twilight as an ally, Twilight was fatally injured by Lilith. Linda convinced Matrix to bond with Twilight to heal her. Twilight/Matrix used her healing powers to restore Linda to her previous power levels. Twilight and Matrix whereabouts are unknown at this time.
Matrix/Linda Danvers was believed to have been retconned out of existence due to Superboy-Prime punching the walls of reality. However, Geoff Johns stated that Linda Danvers still exists, and that she would make her return to the DCU in July of 2008. Linda, not Matrix, had appeared in Reign in Hell, still being considered part angel despite having Matrix leave her before. She has been referenced as Supergirl in the miniseries, thus implying that her series remains in continuity. As far as Matrix is concerned there has been no mention of her and her existence in DC remains uncertain.
Powers & Abilities
Matrix uses psychic abilities to emulate the powers of a Kryptonian. It allows her to have Super strength, speed, and endurance. It allows her to possess flight and makes her invulnerable against most conventional attacks.
Along with her Psychokinesis Matrix has the ability to affect space and matter around her. She can create force fields of psychic energy or she can turn it on the offensive and use psychic blasts against her enemies.
Matrix possesses the ability to render herself invisible by deflecting light waves and sound waves with her aura, she could cloak herself (becoming become invisible and inaudible, even to Superman's hearing)
She could morph into a humanoid form of similar mass, and use her psi-power to rearrange the molecular structure of fabrics, and alter clothing and costumes with her thoughts (an ability that enabled her to change her clothing when she changed shape, and turn tattered rags into a costume for Superman).
After combining herself with Linda Danvers and becoming one of the three Earth Angels, the Earth Angel of Fire, in her case, Matrix lost her invisibility and shape-shifting, but soon learned that she had new powers to compensate for the loss. Her flame abilities most notably manifest as fiery wings and flaming eye beams (similar but to Superman's heat vision, but different in that they are actual flames, and, of course, angelic in origin, rather than being biased on biology apparently).