Properties of matrix addition

The properties of matrix addition are closure property, commutative property of addition, associative property of addition, additive identity property, and additive inverse property. We summarize these properties in the figure below

Properties of matrix addition

The properties of matrix addition along with well chosen examples to illustrate the concept

If A, B, and C are m x n matrices, then

Closure property

A + B is an m x n matrix

Example

Let A = [1 -2 5 3] and let B = [2 0 -4 6]

A is a 1 x 4 matrix and B is also a 1 x 4 matrix.

A + B = [1+2 -2+0 5+-4 3+6]

A + B = [3 -2 1 9]

A + B is also a 1 x 4 matrix

Commutative property of addition

A + B = B + A

Example #1

$$ A = \begin{bmatrix} -2 & 0 & 4\\ 6 & 1 & 3\\ -8 & 1 & 0 \end{bmatrix} B = \begin{bmatrix} 1 & 2 & 0\\ 7 & 9 & 10\\ 0 & 4 & 5 \end{bmatrix} $$

A is a 3 x 3 matrix and B is also a 3 x 3 matrix.

$$ A + B = \begin{bmatrix} -2 + 1 & 0 + 2 & 4 + 0\\ 6 + 7 & 1 + 9 & 3 + 10\\ -8 + 0 & 1 + 4 & 0 + 5 \end{bmatrix} $$
$$ A + B = \begin{bmatrix} -1 & 2 & 4\\ 13 & 10 & 13\\ -8 & 5 & 5 \end{bmatrix} $$
$$ B + A = \begin{bmatrix} 1 + -2 & 2 + 0 & 0 + 4\\ 7 + 6 & 9 + 1 & 10 + 3\\ 0 + -8 & 4 + 1 & 5 + 0 \end{bmatrix} $$
$$ B + A = \begin{bmatrix} -1 & 2 & 4\\ 13 & 10 & 13\\ -8 & 5 & 5 \end{bmatrix} $$

Example #2

Let A = [4   1   -4] and let B = [-4   2   5] 

A is a 1 x 3 matrix and B is also a 1 x 3 matrix.

A + B = [4+-4   1+2   -4+5]

A + B = [0   3   1]

B + A = [-4+4   2+1    5+-4]

B + A = [0   3   1]

Associative property of addition

(A + B) + C  = A + (B + C)

Example

Let A = [2   -1] , B = [0   1], and C = [3   -5] 

A is a 1 x 2 matrix, B is a 1 x 2 matrix, and C is also 1 x 2 matrix.

(A + B) + C = [2+0   -1+1] + [3   -5] 

(A + B) + C = [2   0] + [3   -5] 

(A + B) + C = [2+3   0+-5] 

(A + B) + C = [5   -5] 

A + (B + C) = [2   -1] + [0+3   1+-5] 

A + (B + C) = [2   -1] + [3   -4] 

A + (B + C) = [2+3   -1+-4] 

A + (B + C) = [5   -5]

Additive identity property

There exists a unique m x n matrix O such that A + O = O + A = A

Example #1

Let A = [8 9] and O = [0 0]

A + O = [8 9] + [0 0] = [8+0 9+0] = [8 9] = A

O + A = [0 0] + [8 9] = [0+8 0+9] = [8 9] = A

Example #2

$$ A = \begin{bmatrix} 9 & -5 & 8\\ 1 & 0 & 1\\ 3 & 2 & -6 \end{bmatrix} O = \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} $$
$$ A + O = \begin{bmatrix} 9 + 0 & -5 + 0 & 8 + 0\\ 1 + 0 & 0 + 0 & 1 + 0\\ 3 + 0 & 2 + 0 & -6 + 0 \end{bmatrix} $$
$$ A + O = \begin{bmatrix} 9 & -5 & 8\\ 1 & 0 & 1\\ 3 & 2 & -6 \end{bmatrix} = A $$
$$ O + A = \begin{bmatrix} 0 + 9 & 0 + -5 & 0 + 8\\ 0 + 1 & 0 + 0 & 0 + 1\\ 0 + 3 & 0 + 2 & 0 + -6 \end{bmatrix} $$
$$ O + A = \begin{bmatrix} 9 & -5 & 8\\ 1 & 0 & 1\\ 3 & 2 & -6 \end{bmatrix} = A $$

Additive inverse property

For each A, there exists a unique opposite, -A such that A + (-A) = O 

Example

Let A = [4   -6], -A = [-4   +6]  and O = [0   0]

A + (-A) = [4   -6] + [-4   +6] = [4+-4   -6+6] = [0   0] =  O

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