Matrix Theorems
Here, we list without proof some of the most important rules of matrix algebra 
theorems that govern the way that matrices are added, multiplied,
and otherwise manipulated.
Notation
Matrix Addition and Matrix Multiplication

A + B =
B + A
(Commutative law of addition)

A + B + C =
A + ( B + C ) =
( A + B ) + C
(Associative law of addition)

ABC =
A( BC ) =
( AB )C
(Associative law of multiplication)

A( B + C ) =
AB
+ AC
(Distributive law of matrix algebra)

x( A + B ) =
xA
+ xB
Transposition Rules

( A' )' = A

( A + B )' =
A' + B'

( AB )' =
B'A'

( ABC )' =
C'B'A'
Inverse Rules

AI = IA = A

AA^{1} = A^{1}A = I

( A^{1} )^{1} = A

( AB )^{1} =
B^{1}A^{1}

( ABC )^{1} =
C^{1}B^{1}A^{1}

( A' )^{1} =
( A^{1} )'