Properties of Matrix Operations
The main properties of matrix operations such as addition, multiplication, transpose and inverse are presented.
In what follows, A, B and C are matrices whose sizes are such that the operations are well defined and k is a scalar and n is a positive integer.
In is the identity matrix of size n × n whose diagonal entries are all equal to 1 and all non diagonal entries equal to zero.
0 is the zero matrix whose entries are all zeros.
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- A + 0 = 0 + A = A , where 0 is the zero matrix.
- A + B = B + A Commutativity of Addition of Matrices
- (A + B) + C = A + (B + C) Associativity of Addition of Matrices
- A ( B C ) = ( A B ) C Associativity of Multiplication of Matrices
- A In = A , where In is the identity matrix.
- In A = A
- 0 A = 0 , where 0 is the zero matrix.
- Note that in general AB ≠ BA
Distributive Properties of Matrices
- A(B ~+mn~ C) = AB ~+mn~ AC
- (A ~+mn~ B)C = AC ~+mn~ BC
Properties of Matrix Multiplication by Scalars
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k(A ~+mn~ B) = k A ~+mn~ k B
-
(k ~+mn~ m)A = k A ~+mn~ m A
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k(m A) = (k m)A
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k(A B) = (k A)B = A(k B)
Properties of Matrix Transpose
In what follows, AT is the transpose matrix.
- (AT)T = A
- (A ~+mn~ B )T) = AT ~+mn~ BT
- (k A)T = k AT
- (A B)T = BT AT
- (In )T = In
Properties of Matrix Inverse
In what follows, A-1 is the inverse matrix .
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