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
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
