The properties of matrix addition are closure property, commutative property of addition, associative property of addition, additive identity property, and additive inverse property.
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
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
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
Additive inverse property
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
Jan 12, 22 07:48 AM
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