if x+y+z=0 prove that the determinant   xa  yb  zc              a  b  c
                                                     yc  za   xb  =  xyz  c  a   b
                                                     zb  xc   ya             b  c  a
 
 
 

Asked by ahuja8087 | 5th Sep, 2016, 06:36: PM

Expert Answer:

begin mathsize 16px style To space prove space that colon space open vertical bar table row xa yb zc row yc za xb row zb xc ya end table close vertical bar equals xyz open vertical bar table row straight a straight b straight c row straight c straight a straight b row straight b straight c straight a end table close vertical bar
LHS equals open vertical bar table row xa yb zc row yc za xb row zb xc ya end table close vertical bar
equals xa open vertical bar table row za xb row xc ya end table close vertical bar minus yb open vertical bar table row yc row zb end table table row xb row ya end table close vertical bar plus zc open vertical bar table row yc za row zb xc end table close vertical bar
equals xa left parenthesis yza squared minus straight x squared bc right parenthesis minus yb left parenthesis straight y squared ac minus straight b squared xz right parenthesis plus zc left parenthesis straight c squared xy minus straight z squared ab right parenthesis
equals xyz left parenthesis straight a cubed plus straight b cubed plus straight c cubed right parenthesis minus abc left parenthesis straight x cubed plus straight y cubed plus straight z cubed right parenthesis
equals xyz left parenthesis straight a cubed plus straight b cubed plus straight c cubed right parenthesis minus 3 abc left parenthesis xyz right parenthesis space space space...... left square bracket Since space straight x plus straight y plus straight z equals 0 comma space therefore comma space straight x cubed plus straight y cubed plus straight z cubed equals 3 xyz right square bracket
equals xyz left parenthesis straight a cubed plus straight b cubed plus straight c cubed minus 3 abc right parenthesis
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis left parenthesis straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ca right parenthesis space space space space space....... left parenthesis straight i right parenthesis
RHS equals xyz open vertical bar table row straight a straight b straight c row straight c straight a straight b row straight b straight c straight a end table close vertical bar
straight R subscript 1 rightwards arrow straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3
equals xyz open vertical bar table row cell straight a plus straight b plus straight c end cell cell space space space space space space straight a plus straight b plus straight c space space space space space space end cell cell straight a plus straight b plus straight c end cell row straight c straight a straight b row straight b straight c straight a end table close vertical bar
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis open vertical bar table row 1 cell space space space space space space 1 space space space space space space end cell 1 row straight c straight a straight b row straight b straight c straight a end table close vertical bar
straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 1 comma space space space straight C subscript 3 rightwards arrow straight C subscript 3 minus straight C subscript 1
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis open vertical bar table row 1 cell space space space space space 0 space space space space space end cell 0 row straight c cell straight a minus straight c end cell cell straight b minus straight c end cell row straight b cell straight c minus straight b end cell cell straight a minus straight b end cell end table close vertical bar
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis left square bracket left parenthesis straight a minus straight c right parenthesis left parenthesis straight a minus straight b right parenthesis minus left parenthesis straight c minus straight b right parenthesis left parenthesis straight b minus straight c right parenthesis right square bracket
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis left parenthesis straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ac right parenthesis space space space space space....... left parenthesis ii right parenthesis
From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
LHS equals RHS
Thus comma space open vertical bar table row xa yb zc row yc za xb row zb xc ya end table close vertical bar equals xyz open vertical bar table row straight a straight b straight c row straight c straight a straight b row straight b straight c straight a end table close vertical bar. end style

Answered by Rebecca Fernandes | 5th Sep, 2016, 09:26: PM