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

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

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

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