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Basic Concepts Of Determinant Part 7

Posted on - 04-01-2017

Math

IIT JEE

Linear Equations

  • The system of homogeneous simultaneous linear equations
  • has a non-trivial solution (i.e. at least one of x,y,z, is different from zero ) if
  • If D > 0, then the only solution of the above system of equations is x = 0, y = 0, z = 0 (called trivial solution). .
  • If D = 0, then the number of solutions of the system of equations is infinite (called non-trivial and includes trivial solution).

    Example.1

    If the system of equations 3x + 10 y + 17z = 0, x + 6y + 13z = 0 and 20 x – 13y + l z = 0 has a non-trivial solution then find the solution.

    Solution

    The observation for A.P. property reveals l = -46 to have D = 0 (non- trival solution).

    Let z = k and by first two equations, we have 3x +10 y = - 17 k

    - 3x + 18y = - 39 k, therefore y = - 2k and x = k

    Example.2

    If a, b, c are in G.P. with common ratio r1 and a , b and g are also in G.P. with common ratio r2 then find the condition that r1 and r2 must satisfy in order that the system of equations ax+ ay+z= 0, bx +by+z= 0, cx + gy +z = 0 has only a trivial solution.

    Solution

    As a, b, c are in G. P. with common ratio r1 and a, b, g are in G. P. having common ratio r2, a > 0, a > 0, b= ar1, c= ar12, b = ar2, g = ar22.

    Also the system of equation have only zero( trivial ) solution

    D = = aa

    = aa ( r1 - 1) (r2 - 1) ( r­1 - r2) > 0


    &⇒
    r1 > 1, r2 > 1 and r1 > r2 ( as a, a > 0 )
    . .

     
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