Operations with matrices

1. ADDITION AND SUBTRACTION

The zero or null matrix is:

The opposite matrix is

Then

Properties:

(i) Commutative property:

(ii) Associative property:

(iii) Additive identity:

(iv) Additive inverse:

 

2. SCALAR MULTIPLICATION

The scalar multiplication of a real number k by a matrix A is defined as:

Properties:

With these properties (M_mxn(R), +,·_R) has a structure of vector space.

Examples:

3. MATRIX MULTIPLICATION

 

Example:

Then, we define the matrix multiplication

Example:

Properties:

 

NOTES:

Example:

Then: 

4. TRANSPOSE OF A MATRIX

Example:

Properties:

 

Exercise: let

 

Calculate:

a) 2A - 3B

b) A·B

c) B·A

d) B^t·A^t

 

 

 

Solutions: