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

A, B, and C are matrices.
A' is the transpose of matrix A.
A^-1 is the inverse of matrix A.
I is the identity matrix.
x is a real number.

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^-1A = I

( A^-1 )^-1 = A

( AB )^-1 = B^-1A^-1

( ABC )^-1 = C^-1B^-1A^-1

( A' )^-1 = ( A^-1 )'