Moderate# Sum of Invertible not Invertible

LINALG-BV3XOK

Let $M_2(\mathbb{R})$ be the vector space consisting of all real $2\times 2$ matrices where addition means matrix addition and scalar multiplication means multiplying every element of the matrix by that scalar.

Define $GL_2(\mathbb{R})$ to be the subset of $M_2(\mathbb{R})$ consisting of those matrices which are invertible.