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

## 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

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]

(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]

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

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

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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