Properties of matrix addition

Properties of matrix addition

In this section, we will look at the properties of matrix addition. These properties include the dimension property for addition, commutative property, and associative property. Note that for the dimension property, we are allowed to add or subtract two matrices with the same dimensions. The commutative property states that changing the order of the addition or subtraction of two matrices lead to the same result. For the associative property, changing what matrices you add or subtract one will lead to the same answer. There are also matrix addition properties for identity and zero matrices as well. Adding a zero matrix with another matrix (call it A) will give back A. Multiplying an identity matrix with another matrix (call it B) will give back B.


Let the matrices X,YX,Y and ZZ have equal dimensions. Then we have the following matrix addition properties:

Dimension property for addition
If XX and YY has the same dimensions, then X+YX+Y also has the same dimensions.
Commutative property
Associative property

There are also some matrix addition properties with the identity and zero matrix.

Property for the zero matrix
There is always a zero matrix OO such that O+X=XO+X=X for any matrix XX.
Property for the identity matrix
Let XX be a nn by nn matrix. Then there is an identity matrix InI_n such that InX=XI_n \cdot X=X.
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Properties of matrix addition

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