Distributive law 
Left side A(B + C) = AB + AC
Right side (A + B)C = AC + BC

A(m ✕ n) B and C (n ✕ p)
A and B (m ✕ n) C(n ✕ p)

Associative law 
Addition (A + B) + C = A + (B + C)
Multiplication (AB)C = A(BC)

(m ✕ n)
(n ✕ n)

Scalar multiplication 
(kA)B = A(kB) = k(AB) 
A(m ✕ n) B(n ✕ p) k any number 
Commutative law 
Addition A + B = B + A
Multiplication not commutative

A and B (m ✕ n)
Because A∙B ≠ B∙A

Other algebric laws
(k, v are constants) 
0 + A = A 
k(A + B)= kA + kB 
1 · A = A 
(k + v)A = kA + vA 
0 · A = 0 
k(vA) = (kv)A 
A + ( A) = 0 
kA = 0 → k = 0 or A = 0 
( 1)A = A 


Matrices powers
(c is constant) 
