Matrix Addition and Subtraction

For two matrices and of the same size, the matrix formed by adding the corresponding elements of and is called the sum of and , denoted by . Similarly, the matrix formed by subtracting each element of from the corresponding element of is called the difference of and , denoted by .

For matrices and :

1. 
   
   

Example 1. Calculate the following expression.

 Solution

Example 2. Calculate the following expression.

 Solution

Properties of Matrix Addition

Matrix addition has the following properties:

1. Commutative Property:
2. Associative Property:

Zero Matrix

1. A matrix where all elements are zero is called a zero matrix, denoted by . For example:
   ,
   
   ,
   
   
   These are zero matrices of sizes , , and respectively.
   
   
2. For any matrix , the zero matrix of the same size satisfies:

Negative of a Matrix ()

1. The matrix formed by changing the sign of every element of matrix is called . For example, if
   , then
   
   
2. For any matrix and a zero matrix of the same size:
   
   
3. For two matrices and of the same size:

Scalar Multiplication of a Matrix

For any real number , the matrix formed by multiplying each element of matrix by is called the scalar multiple of by , denoted by .
For matrix :

Properties of Scalar Multiplication

For two matrices and of the same size, and real numbers and :

1. , , , (where is a zero matrix of the same size).
2. Associative Property:
3. Distributive Properties:
   
   

Example. Given , find .

 Solution

If you multiply each element of matrix by , then: