Integration of Determinants

In mathematics, a determinant refers to a square matrix consisting of functions of entries whose values are scalar. It consists of rows and columns, say n m. By multiplying the determinant elements in the rows and columns, we will get a single simplified value, which is also a scalar quantity. In some cases, the rows and columns will be equal, which is nn and in others they may vary which is denoted by nm.

It is denoted by det (variable) say A, det(A), A or (A).

Where determinant value comes as (ad)-(cb)    

The above is a 22 determinant, whose final value is 6.

Ie., (64)-(92)

      24-18 = 6

How to solve a determinant:

      (x)=

(x)= a(nr-qo)-b(mr-po)+c(mq-pn)

(y)=

(y)=3(5-4)-5(10-1)+4(8-1)

=3(1)-5(9)+4(7) = 3-45+28

=-14

Thus, a 33 determinant is obtained in such a way.

Terms used in Integration of Determinants: 

We generally use the term elements for the numerical values or the variables inside the variable. Minors and cofactors are other terms generally used in determinants.

Minor is obtained by eliminating or crossing out the row and the column in which the element is present. Each element in the determinant has its minor. Its value is obtained by taking out the determinant by eliminating one or more rows and columns. It is available for nm determinant.

Cofactor is given by the Minor element with definite representation. It is a signed minor. Similar to the minor, cofactors are available are nm matrix.

Integration of determinants notes:

In mathematics, Integral refers to assigning of values to the function, which refers to various concepts. The process of finding these integrals is known as Integration.

In an integral calculus, we apply the values of limits and find out the values – the lower and upper limits. The limits are applied such that the lower limit is subtracted from the upper limit.

If the value of these limits are finite and gives a constant output, it is known as definite integral and when it is not constant say , it  is indefinite integration.

Problem solving method:

Let us take a determinant x of 33

Let (x)=

On integrating the above 33 |det|, whose limits are given by ab,then

ab (x)dx=  

Here  f(x),g(x),h(x) are functions of x

          m, n, o, p, q and r are constants

(x)=f(x)(nr-qo)-g(x)(mr-po)+h(x)(mq-pn)

ab(x)dx=abf(x)(nr-qo)-g(x)(mr-po)+h(x)(mq-pn)

              =(nr-qo)abf(x)dx-(mr-po)abg(x)dx+(mq-pn)abh(x)dx

If the elements of function x are present in more than 1 column or row, the integration value is obtained by the expansion of the determinant.

Examples of Integration 

If (x)= 

then find the value of 01(x)dx

Solution :

(x)=

01(x)dx=x2((5x+5)-0)-6x((x+1)-0)+x(3-(-5))

  =x2(5x+5)-6x(x+1)+x(3+5)

               =5x3+5x2-6x2-6+8x

               =5x3–x2+8x-6

01(x)dx=01[5x44–x33+8x22-6x]+c

Where C   is the Integration Constant.

Substituting the limit values,we will get

               =54–13+4-6

               =54–13-2

               =-1312

Example 2:

Find xdx for given determinant whose limits are 0/4

Solution :

(x)=

0/4xdx = 

=

=-1+4+52

=–112

Conclusion

We learnt nm determinants, how to arrive at the single final solution for that determinant. The concept of integrating  the determinants was discussed above. Now, we can solve any kind of questions in Determinants Integration.