## Operations on Determinants

Multiplication of two DeterminantsTwo 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, R

_{1}→ first row of first determinantRi → first row of second determinant

Illustration:Find the product of the determinants

Solution:=

Differentiation of a DeterminantYes, 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 DeterminantsLet 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 C

_{1}of all determinants Δ_{a}we have

By taking (n-1)n/2 as common factor from C

_{1}and 6 as common factor from C_{3}, we get

Since C

_{1}and C_{3}are identical ∑^{n}_{a=1}∆_{a}=0.

Special Determinants

1. Symmetric determinantThe elements situated at equal distance from the diagonal are equal both in magnitude and sign.

= abc + 2fgh - af

^{2}- bg^{2}- ch^{2}.

2. Skew symmetric determinantAll 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.

= -(a

^{3}+ b^{3}+ c^{3}- 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).

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