Operations on Determinants

  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 1st, 2nd & 3rd rows of other determinant. The three expressions thus obtained will be elements of 1st row of resultant determinant. In a similar manner the element of 2nd & 3rd rows of determinant are obtained.

         multiplication-of-two-determinants

where, R1 → first row of first determinant

          Ri  → first row of second determinant

Illustration:

        Find the product of the determinants

        matrix58

 Solution:

         = matrix59

Differentiation of a Determinant

        Yes, let f(x) = matrix60 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)

        = matrix61

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'.

        determinant-of-order-n

Summation of Determinants

Let Δr=matrix62, where a,b,c,l,m and n are constants independents of r,

then  =∑nr=1  ∆rmatrix63

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 Δa=matrix64. Showthat ∑na=1a = 0   

Solution:

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

       matrix65 

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

         matrix66

 Since C1 and C3 are identical ∑na=1  ∆a =0.

Special Determinants

1. Symmetric determinant

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

       matrix67 = 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.

         matrix68  = 0.

3.      Circulant determinant:

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

       matrix69 = -(a3 + b3 + c3 - 3abc).

4.   matrix70 = (a-b)(b-c)(c-a).

5.   matrix71 = (a-b)(b-c)(c-a)(a+b+c).

6.   matrix72 = (a-b)(b-c)(c-a)(ab+bc+ca).

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