`text(Property-1 :)`

The value of a determinant remains unaltered, if the rows & columns are interchanged. e.g. if

`D=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=|(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|=D'`

`D` & `D'` are transpose of each other. If `D'=-D` then it is Skew symmetric determinant

but `D' = D => 2 D = 0 => D = 0 =>` Skew symmetric determinant of third order has the value zero.

`text(Remember:)` Without expanding prove that the value of the determinant

`D=|(0,b,-c),(-b,0,a),(c,-a,0)|=0`

`text(Note :)` The value of a Skew symmetric determinant of odd order is zero.


`text(Property-2 :)`

If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g.

Let `D=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|` & `D'=|(a_2,b_2,c_2),(a_1,b_1,c_1),(a_3,b_3,c_3)|`

Then `D'=-D`.


`text(Property-3 :)`

If a determinant has any two rows (or columns) identical , then its value is zero.

e.g. Let `D=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|` then it can be verified that `D=0`.


`text(Property-4 :)`

If all the elements of any row (or column) be multiplied by the same number, then the determinant is multiplied by that number.

e.g. If `D=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|` and `D'=|(Ka_1,Kb_1,Kc_1),(Ka_2,Kb_2,Kc_2),(Ka_3,Kb_3,Kc_3)|` Then `D'=KD`


`text(Property-5 :)`

If each element of any row (or column) can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants. e.g.

`|(a_1+x,b_1+y,c_1+z),(a_2,b_2,c_2),(a_3,b_3,c_3)|=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|+|(x,y,z),(a_2,b_2,c_2),(a_3,b_3,c_3)|`


`text(Property-6 :)`

The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column).

e.g. Let `D=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|` and `D'=|(a_1+ma_2,b_1+mb_2, c_1+mc_2),(a_2,b_2,c_2),(a_3+na_1, b_3+nb_1,c_3+nc_1)|`.

Then `D'=D`.

Note : that while applying this property atleast one row (or column) must remain unchanged.


`text(Property-7 :)`

If by putting `x = a` the value of a determinant vanishes then `(x - a)` is a factor of the determinant.



`text(Property-8 :)`

ln a determinant the sum of the product's of the element's of any row (column) with their corresponding co factors is equal to the value of determinant

Let `D=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|`

Let `A_i,B_i,C_i` be the cofactor's of the element's `a_i, b_i, c_i (i=1,2,3)`

Then `a_1A_1 + b_1B_1 + c_1C_1 = D`

`a_2A_2 + b_2B_2 + c_2C_2 = D`

Similarly,

ln a determinant the sum of the product's of the element's of any row( column) with the cofactor's of corresponding element's of any other row ( colunm) is zero.

i.e. `a_1A_2 + b_1B_2 + c_1C_2 = 0` or `a_2A_1 + b_2B_1 + c_2C_1 = 0`.

`text(Important Results : )`

1. If any row (or column) of a determinant `Delta` be passed over `m` rows (or columns), then the resulting determinant `= (-1)^mDelta.`

2. If each element of first row of a determinant consists of algebraic sum of p elements, second row consists of algebraic sum of q elements, third row consists of algebraic sum of r elements and so on.
Then, given determinant is equivalent to the sum of `p xx q xx r xx ...` other determinants in each of which the elements consists of single term.

`text(Remember:)`

Factorisation in respect the following determinants are very useful and should be remembered.