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 )'