Determinants.

 Determinant is an array type, but this must have the same number of rows and the same number of columns which is called a square matrix. Therein do not apply the four operations, but has its properties, how to find the numerical value of a determinant.


Determinant of 1st order.
    Given a square matrix of 1st-order M = [a11], the determinant is the actual number a11:

  det = M = a11 Ia11I

Note: We represent the determinant of a matrix between two vertical bars, which have no meaning module.

    For example:  
      ◘ M= [5] --> det M = 5 ou I 5 I = 5                   ◘ M = [-3] --> det M = -3 ou I -3 I = -3




Determinant of 2nd order.
    Given the matrix of order 2, by definition the determinant associated to M, 2nd order determinant is given by:

     Therefore, the determinant of a matrix of order 2 is given by the difference between the product of the diagonal elements and the product of the secondary diagonal elements. See the following example.

  

                       

Determinant of 3rd order.

The calculation of the determinant of the 3rd order can be done by way of a practical device, called Sarrus rule.

    Watch how we apply this rule to..


Step 1: repeat the first two columns next to the third:

Step 2: We find the sum of the product of the diagonal elements with the two products obtained by multiplying the elements of parallel to that diagonal (the sum should be preceded by the plus sign):

Step 3: We find the sum of the product of the diagonal elements with the secondary two products obtained by multiplying the elements of parallel to that diagonal (the sum should be preceded by the minus sign):

thus:

Note: If we develop this determinant 3rd order applying the Laplace theorem, we find the same real number.