Lying in the intersection of ring theory and linear algebra, this work focuses on "invertible algebras," algebras in which every element can be uniquely written as a sum of invertible elements, and on "I2 algebras," invertible algebras where each element can additionally be written as a sum of the inverses of the original elements. We present conditions on algebras to determine if they are invertible or I2, and explore the connection between these notions.