Determinants

A determinant of order two is written as   where and the value of this determinant will be .

In the same manner determinant of order three and four 

Horizontal lines are called rows and vertical lines are called columns and in any determinant are called diagonals elements.

All the elements in the determinant   are complex number.

Note** - All the real number are also a complex number.

If all the elements in the determinant is a real number then that is called the real determinant.

If at least one number is Imaginary number in the determinant then that is known as the imaginary determinant.

Minors:-  If we delete row and column passing through the element  , the determinant thus obtain is called minor and is usually define by  .

Like we have determinant 

, then the minor of  i.e.

 Note that 1st column and 2nd row are deleted.

Cofactors:-  is called cofactor of   and it is denoted by  .

Properties of the determinants 

1. The value of determinant does not get change when we change the row into column and column into a row. The determinant obtain by interchange by row and column is called transpose of the determinant and it is denoted by   .

Thus by the above properties  

2. If all the element of row or column is zero then the , then determinant is zero .

3. When we change any two  row or any two-column of the determinant then the sign of determinant will get change.

If  row gets interchange then . 

4. If all the elements of the row or column get to multiply by any constant number then value of determinant also gets multiplied by that constant.

If  condition   .

5. If all the element of any row or column are proportional or identical to other row or column then determinant must be zero.

Example  

6. If each element of any row or column is sum of two numbers, the determinant can be expressed as the sum of determinants of the same order.

Example  

It should be remain same in the both determinant of right hand side.

We can also express this in three number sum determinants.

7. If any determinant  become zero on putting x=a then (x – a) is a factor of .

8. Determinant which all elements are zero except diagonals elements, is equal to the product of diagonal element.

Example:-   and also  is called unit determinant and its value id 1.

9. If all the element of a determinant below the diagonal or above the diagonal are zero , then that is called triangular determinant.

Example :  

10. Skew-Symmetric determinants:- if the determinant then the determinant is called skew-symmetric determinants.

The value of odd order Skew-Symmetric determinants is always zero.

Example  

Product of two Determinants:- 

 Let then 

Multiplication also perform by row by column or column by row as require in the problems.

Determinants of the Co-factors :-  

Let  denotes the determinant of the co-factors that   where   and if determinants of order 3 and  then 

System Of Linear Equations :- 

1. Homogeneous linear equation:- 

Consider the system of homogeneous linear equation in three unknowns x, y and z

Clearly this system always has a solution x = y = z = 0, which is called zero solution or trivial solution.

Case I : If  there is only one solution x = y = z = 0 

Case II : If  the system has infinite solutions and so non-trivial solution.

 2. Cramer’s Rule 

Consider the system of linear equation 

We define  and  which is obtain by supressing the column of the coefficient of x and replacing it by a column of the constant term on right-hand side.

Similarly   and 

Now 

Case I : if   , solution of system ( 1 ) is given by, , and the system is called consistent. 

Case II : but at least one of   then the system does not possess any common solution and so system is called inconsistent.

Case III : (i) : and also and at least one co-factor of  , then the system have infinite many solution and the system can be solved by elimination method.

(ii) : if  and all cofactor are zero then the system is equivalent to only one equation in three unknowns and then we given any two unknowns arbitrary values and find the remaining unknowns in term of three constant. 

SOME FUNDAMENTAL DETERMINANTS 

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