# Linear Algebra

## Commutative Law

Similar to the addition of algebraic expressions or vectors, commutative law allows the addition of matrices without dependence on order. If $$A, B \in M_{m \times n} ( \mathbb{R})$$, we have that:

$$A + B = B + A$$

Consider $$A = \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix}$$ and $$B = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}$$ (Recall addition of matrices from Operations on Matrices.)

 $$A + B$$ = $$\begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} + \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}$$             = $$\begin{bmatrix} 2+3 & -5+5 \\ -1+1 & 3+2 \end{bmatrix}$$             = $$\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$$ $$B + A$$ = $$\begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} + \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix}$$                  $$\begin{bmatrix} 3+2 & 5+(-5) \\ 1+(-1) & 2+3 \end{bmatrix}$$                  $$\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$$

We see that $$A + B = B + A$$

## Associative Laws

When dealing with matrices, we can apply the associative laws of addition and multiplication. This means that if we have $$A, B, C \in M_{m \times n} (\mathbb{R})$$ then:

$$1)$$ $$A + (B + C) = (A + B) + C$$

$$2)$$ $$A(BC) = (AB)C$$

 Note: $$AB \neq BA$$ and $$A(BC) \neq (BC)A$$. Matrix multiplication is not commutative!

## Distributive Law

If we are dealing with matrix multiplication combined with matrix addition, we can apply the distributive law. If $$A, B, C \in M_{m \times n} ( \mathbb{R})$$ then:

$$A(B + C) = AB + AC$$