# 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 $k$ and $m$ are scalars.
$I_n$ is the identity matrix of size $n \times n$ 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.

1.    $A + 0 = 0 + A = A$ , where $0$ is the zero matrix.
2.    $A + B = B + A$
3.    $(A + B) + C = A + (B + C)$

## Properties of Matrix Multiplication

1.    $(A B) C = B(A C)$
2.    $A I_n = A$ , where $I_n$ is the identity matrix.
3.    $I_n A = A$
4.    $0 A = 0$ , where $0$ is the zero matrix.
5.    In general $AB \ne BA$

## Distributive Properties of Matrices

1.    $A(B \pm C) = AB \pm AC$
2.    $(A \pm B)C = AC \pm BC$

## Properties of Matrix Multiplication by Scalars

1.    $k(A \pm B) = k A \pm k B$
2.    $(k \pm m)A = kA \pm mA$
3.    $k(m A) = (k m)A$
4.    $k(AB) = (kA)B = A(kB)$

## Properties of Matrix Transpose

1.    $(A^T)^T = A$
2.    $(A \pm B)^T = A^T \pm B^T$
3.    $( k A)^T = k A^T$
4.    $(A B)^T = B^T A^T$
5.    $(I_n)^T = I_n$

## Properties of Matrix Inverse

1.    $A A^{-1} = A A^{-1} = I_n$
2.    $(A^{-1})^{-1} = A$
3.    $(k A)^{-1} = k^{-1} A^{-1} = \dfrac{1}{k} A^{-1}$ , for $k \ne 0$
4.    $(A B)^{-1} = B^{-1} A^{-1}$
5.    $I_n^{-1} = I_n$
6.    $(A^T)^{-1} = (A^{-1})^T$