Sum of matrices

Matrix addition is a linear operation that consists of unifying the elements of two or more matrices that coincide in position within their respective matrices and that they have the same order.

In other words, the sum of one or more matrices is the union of the elements that have the same position within the matrices and that they have the same order.

Matrix addition formula 

Screenshot 2019 07 30 A Les 18.31.44
Sum of multiple matrices of the same order.

Process

To add matrices we must: 

  1. Check the order of the matrices, such that: 
    • If the order of the matrices is the same , then the matrices can be added. 
    • If the order of the matrices is different , then  we can not add the matrices. 
  2. Add the elements that have the same position within their respective matrices.

Matrix summation shares the same characteristics as when adding numbers and variables in algebra, with the difference that here we have “coordinates”. That is, we will take into account the position of the element within each matrix. The position of each element is denoted with subscripts, such that: 

Screenshot 2019 07 30 A Les 18.29.46
Sum of elements with the same position in their respective matrices.

So, the summation of these three elements is possible since they all have the same position. In other words, they have the same numbers in the subscripts. 

If the position of the elements were different, we could not add them. 

Screenshot 2019 07 30 A Les 18.30.37
We cannot add elements of different positions in their respective matrices.

Matrix sum properties

Given any three matrices X, Z, Y such that: 

Screenshot 2019 07 30 A Les 18.40.45
Matrices of order nxm.
  • Associative property: 

Z + (X + Y) = (Z + X) + Y

It is equivalent to first add two matrices and then another matrix to the previous result. 

  • Commutative property: 

Z + X + Y = X + Y + Z

The order of the summation is not relevant. 

  • Neutral element:

Given a zero matrix O of the same order as Z, X, Y, such that:

Screenshot 2019 07 30 A Les 18.42.27
Zero or null matrix.

So, 

X + O = O + X = X

The neutral effect occurs when we add the target matrix to a zero matrix. The result is the same matrix. 

  • Distributive property: 

(X + Z) h = X h + Z h

Unlike matrices, powers that do not meet the distributive property in the sum. 

General example

Sum of two square matrices of order 2:

Screenshot 2019 07 30 A Les 18.37.06
Sum of square matrices of order 2.

Sum of two square matrices of order 3:

Screenshot 2019 07 30 A Les 18.37.37
Sum of square matrices of order 3.

Question

To add matrices, should we check that they are square matrices?

To add two or more matrices we first have to check that they have the same order. We can add any matrix with other matrices, even if they are not square matrices.  

Theoretical example

Given the matrices Z, X, Y: 

Screenshot 2019 07 30 A Les 18.45.09
Matrices of different order.

We add:

Screenshot 2019 07 30 A Les 18.45.34

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