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Linear Algebra

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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.

A

$GL_2(\mathbb{R})$ is a subspace of $M_2(\mathbb{R})$ because the product of two invertible matrices is invertible

B

$GL_2(\mathbb{R})$ is a subspace of $M_2(\mathbb{R})$ because the sum of two invertible matrices is invertible

C

$GL_2(\mathbb{R})$ is not a subspace of $M_2(\mathbb{R})$ because the sum of two invertible matrices is not necessarily invertible

D

$GL_2(\mathbb{R})$ is not a subspace of $M_2(\mathbb{R})$ because the inverse of an element of $GL_2(\mathbb{R})$ is not necessarily contained in $GL_2(\mathbb{R})$