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-1A = I

  • ( A-1 )-1 = A

  • ( AB )-1 = B-1A-1

  • ( ABC )-1 = C-1B-1A-1

  • ( A' )-1 = ( A-1 )'