__Properties of determinants__ (Secondary Information) :-

1. If rows be changed into columns and columns into the rows, the determinant remain unaltered .

2. If any two rows or columns of a determinant are interchanged the resulting determinant is the negative of the original determinant .

3. If two rows (or ltwo columns) in a determinant have corresponding entries that are equal, the value of the determinant K, then determinant is multiplied by K .

4. If each of the entries in a row (or column) of a determinant is multiplied by K, then determinant is multiplied by K .

5. If each entry in a row (or column) of a determinant is written as the sum of two or more trems than the determinant can be written as the sum of two more determinants .

6. If to each element of a line (row or column) of a determinant be adeled the equimultiples of the corresponding elements of one or more parallel lines, the determinant remains unaltered .

7. If each entry in any cow (or column) of determinant is zero, then the volue of detrminant equal to zero .

8. If determinant varishes for X = a, then (X - a) is a factor of D .

__Product of two Determinants__ :-

__Differentiation of a determinant__ (Sexondary Information) :-

(X) =

then

(X) =

Similarly if

(X) =

, then

^{1} (X) =

__Summation of Determinants__ :-

(Secondary Information)

Let

r =

Then

__Integration of Determinants__ (Secondary Information ) :-

(X) =

__Special Determinants__ (Secondary Information) :-

1. Symmetric Determinant :-

The elements situted at equal distance from the diagonal are equal both in magnitude and sign.

eg.