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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 :





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:
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