# A calculus of the absurd

##### 22.1.3 Properties of matrix addition

For all $$m \times n$$ matrices $$A$$ and $$B$$, there exists a value $$X$$ such that

$$A + X = B$$

• Property 22.1.1

For all values of $$m,n$$ there exists an $$m \times n$$ matrix $$0$$, such that all $$m \times n$$ matrices $$A$$ satisfy the property

$$A + 0 = 0 + A$$

• Property 22.1.2 It is also the case that for any $$m \times n$$ matrix $$A$$, there exists a matrix $$-A$$, such that

$$A + (-A) = 0$$

• Property 22.1.3 For all $$n \times n$$ matrices $$A$$ and $$B$$, it is the case that

$$A + B = B + A$$

Together, all these properties mean that $$\mathbb {R}^{m \times n}$$ forms an Abelian Group under multiplication.