0

Guest

Courses New
Book a free demo of live class

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  ?r = matrix63

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=1 ?a = 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).

Be a participant in the online tests conducted by askIITians and also give answers in AQAD ( A Question  A Day) get a number of benefits in the online live classroom courses and also read online Study material for IIT JEE and JEE Main /Advanced (known as AIEEE till 2012) preparation absolutely free.

To read more, Buy study materials of Matrices and Determinants comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.


TOP Your EXAMS!

Upto 50% Scholarship on Live Classes

Course Features

  • Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution

Book a Live class Live courses Ask question
r