Determinants

 

 The concept of derminantwas disussed by the German Mathematician G.W. Leibnitz (1676-1714) and Cramer (1750) developed the rule for solving linear equation using determinants.

Value of Determinant :

      A determinant is a scalar value that can be calculated from the elements of a square matrix. 

The representation is defined as the determinant of order two. Number a, b, c, d are called elememts of the determinant . In this arrangement , there are two rows and columns. The value of determinant  is  ad - bc .

Determinant of Order 3 :

       A derminant of order 3 is a square arrangement of 9 elements enclosed between two vertical bars. The elements are arranged in 3 rows and 3 column as given below.

Here a_ij represente the elelment in i^th row and j^thcolumn of the determinant.

eg., a₃₁ represents the element in 3rd row and 1st column .

In general , we denote determinant by Capital letter or by Δ(delta).

we can write the rows and columns separatly . eg. The 2nd row is [a₂₁  a₂₂ a₂₃  ]  and 3rd column is 

Expansion Of Determinant :

     There are six ways of expanding a derminant of order 3, corresponding to each of three rows       (R₁ , R₂ , R₃ ) and three columns (C₁ ,C₂ , C₁ ). 

   

The derminant can be expanded as follows :

 

Minors and Cofactors of elements of Determinants :

Let 

   be a given determinant.

The minor of  a_{ij  :}

       It is defined as the determinant obtained by eliminating the ith  row and jth column of A . That is the row and the column that contain the element  a_ij are omitted. We denote the minor of the element  a_ij by  M_{ij .}

Similarly we can find minos of other elements .

Cofactor of  a_ij :

     Cofactor of  a_ij = (-1)^i+j minor of   a_ij  =  C_ij 

∴ Cofactor of element  a_ij = C_ij  = (-1)^i+j  M_ij 

The same definition can also be given for elements in 2☓2 determinant . Thus in 

 The minor of a is d.

The minor of b is c.

The minor of c is b.

The minor of d is a.

Properties Of Derminants :

1. The value of derminant remains unchanged if its rows are turned into columns are turns into rows.

2. If any two rows (or columns) of the determinant are interchanged then the value of determinant              changed its sign. 

The operation  R_i ↔ R_j   changed  the sign of the determinant.

3. If any two rows (or column) of a determinant are identical then the value of determinant is zero .

4. If each element of a row (or a column) of determinant is multiplied by a constant k then the value of the new determinant is k times the value of given determinant.

The operation   R_i → kR_i gives multiple of the determinant by k.

5. If each element of a row(or a column) is expressed as the sum of two numbers then the determinant can be expressed as sum of two determinants.

for example.,

6. If a constsnt multiple of all elememts of any row(or column) is added to the corresponding elements of any other row (or column) then the value of new determinant so obtained is the same as that of the original determinant . The operation R_i ↔ R_i + kR_j   does not changed the value of the determinant.

R_i ↔ R_i + kR_j

 

7.  If each element of a determinant above or below the main diagonal is zero then the value of the seterminant is equal to product of it's diagonal elements. 

That is ,