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]

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

\$\$

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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