# 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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1. A + 0 = 0 + A = A , where 0 is the zero matrix.
2. A + B = B + A Commutativity of Addition of Matrices
3. (A + B) + C = A + (B + C) Associativity of Addition of Matrices

## Properties of Matrix Multiplication

1. A ( B C ) = ( A B ) C     Associativity of Multiplication of Matrices
2. A In = A , where In is the identity matrix.
3. In A = A
4. 0 A = 0 , where 0 is the zero matrix.
5. Note that in general   AB ≠ BA

## Distributive Properties of Matrices

1. A(B ~+mn~ C) = AB ~+mn~ AC
2. (A ~+mn~ B)C = AC ~+mn~ BC

## Properties of Matrix Multiplication by Scalars

1. k(A ~+mn~ B) = k A ~+mn~ k B
2. (k ~+mn~ m)A = k A ~+mn~ m A
3. k(m A) = (k m)A
4. k(A B) = (k A)B = A(k B)

## Properties of Matrix Transpose

In what follows, AT is the transpose matrix.

1. (AT)T = A
2. (A ~+mn~ B )T) = AT ~+mn~ BT
3. (k A)T = k AT
4. (A B)T = BT AT
5. (In )T = In

## Properties of Matrix Inverse

In what follows, A-1 is the inverse matrix .