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.
- 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
- ( A' )' = A
- ( A + B )' = A' + B'
- ( AB )' = B'A'
- ( ABC )' = C'B'A'
- 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 )'