Multiplication of two Determinants

        Two determinants can be multiplied together only if they are of same order. The rule of multiplication is as under:

        Take the first row of determinant and multiply it successively with 1^st, 2^nd & 3^rd rows of other determinant. The three expressions thus obtained will be elements of 1^st row of resultant determinant. In a similar manner the element of 2^nd & 3^rd rows of determinant are obtained.

        where, R₁ → first row of first determinant

                  Ri  → first row of second determinant

Illustration:

        Find the product of the determinants

 Solution:

         =

Differentiation of a Determinant

        Yes, let f(x) = be a given function.

        => f(x) = a(x) d(x) - c(x) b(x)

        => f'(x) = a'(x) d(x) + a(x) d'(x) - c'(x) b(x) - c(x) b'(x)

        => a'(x) d(x) - c'(x) b(x) + a(x) d'(x) - c(x) b'(x)

        =

        Thus, the differential coefficient of a determinant is obtained by differentiating a single row (or column) at a time and finally adding the determinants so obtained.

        Thus, for a determinant of order 'n'.

       

Summation of Determinants

Let Δ_r=, where a,b,c,l,m and n are constants independents of r,

then  =∑ⁿ_r=1  ∆_r =

Here functions of r can be the elements of only one row or column. None of the elements other of than that row or column should be dependent on r.

Illustration:

       Let Δ_a=. Showthat ∑ⁿ_a=1 ∆_a = 0   

Solution:

By adding all the corresponding elements of C₁ of all determinants Δ_a we have

By taking (n-1)n/2 as common factor from C₁ and 6 as common factor from C₃, we get

Since C₁ and C₃ are identical ∑ⁿ_a=1  ∆_a =0.

Special Determinants

1.      Symmetric determinant

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

       = abc + 2fgh - af² - bg² - ch².

2.    Skew symmetric determinant

All the diagonal elements are zero and the elements situated at equal distance form the diagonal are equal in magnitude but opposite in sign. The value of a skew symmetric determinant of odd order is zero.

           = 0.

3.      Circulant determinant:

The elements of the rows (or columns) are in cyclic arrangement.

       = -(a³ + b³ + c³ - 3abc).

4.    = (a-b)(b-c)(c-a).

5.    = (a-b)(b-c)(c-a)(a+b+c).

6.    = (a-b)(b-c)(c-a)(ab+bc+ca).