The properties of matrix addition are closure property, commutative property of addition, associative property of addition, additive identification property, and additive inverse property. We summarize these properties within the determine beneath

The properties of matrix addition together with effectively chosen examples for example the idea

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

Closure property

A + B is an m x n matrix

Instance

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

A is a 1 x 4 matrix and B can be a 1 x 4 matrix.

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

A + B = [3 -2 1 9]

A + B can be a 1 x 4 matrix

Commutative property of addition

A + B = B + A

Instance #1

$$
A = start{bmatrix}
-2 & 0 & 4
6 & 1 & 3

-8 & 1 & 0
finish{bmatrix}
B =
start{bmatrix}
1 & 2 & 0
7 & 9 & 10

0 & 4 & 5
finish{bmatrix}

$$

A is a 3 x 3 matrix and B can be a 3 x 3 matrix.

$$
A + B = start{bmatrix}
-2 + 1 & 0 + 2 & 4 + 0
6 + 7 & 1 + 9 & 3 + 10

-8 + 0 & 1 + 4 & 0 + 5
finish{bmatrix}

$$

$$
A + B = start{bmatrix}
-1 & 2 & 4
13 & 10 & 13

-8 & 5 & 5
finish{bmatrix}

$$

$$
B + A = start{bmatrix}
1 + -2 & 2 + 0 & 0 + 4
7 + 6 & 9 + 1 & 10 + 3

0 + -8 & 4 + 1 & 5 + 0
finish{bmatrix}

$$

$$
B + A = start{bmatrix}
-1 & 2 & 4
13 & 10 & 13

-8 & 5 & 5
finish{bmatrix}

$$

Instance #2

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

A is a 1 x 3 matrix and B can be 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)

Instance

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 can be 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 identification property

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

Instance #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

Instance #2

$$
A = start{bmatrix}
9 & -5 & 8
1 & 0 & 1

3 & 2 & -6
finish{bmatrix}
O =
start{bmatrix}
0 & 0 & 0
0 & 0 & 0

0 & 0 & 0
finish{bmatrix}

$$

$$
A + O = start{bmatrix}
9 + 0 & -5 + 0 & 8 + 0
1 + 0 & 0 + 0 & 1 + 0

3 + 0 & 2 + 0 & -6 + 0
finish{bmatrix}

$$

$$
A + O = start{bmatrix}
9 & -5 & 8
1 & 0 & 1

3 & 2 & -6
finish{bmatrix}

= A

$$

$$
O + A = start{bmatrix}
0 + 9 & 0 + -5 & 0 + 8
0 + 1 & 0 + 0 & 0 + 1

0 + 3 & 0 + 2 & 0 + -6
finish{bmatrix}

$$

$$
O + A = start{bmatrix}
9 & -5 & 8
1 & 0 & 1

3 & 2 & -6
finish{bmatrix}

= A

$$

Additive inverse property

For every A, there exists a singular reverse, -A such that A + (-A) = O 

Instance

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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