Using properties of determinats, prove the following:
$open vertical bar table row straight a cell space straight b end cell cell space space straight c end cell row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals space straight a cubed plus straight b cubed plus straight c cubed minus 3 abc.$

Solution

L.H.S. = $open vertical bar table row straight a cell space straight b end cell cell space space straight c end cell row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals space open vertical bar table row cell straight a plus straight b plus straight c end cell cell space space space space straight a plus straight b plus straight c end cell cell space space straight a plus straight b plus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell space straight c minus straight a end cell row cell straight b plus straight c end cell cell straight c plus straight a end cell cell straight a plus straight b end cell end table close vertical bar comma space by space straight R subscript 1 plus straight R subscript 3$
$equals space left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 1 1 row cell straight a minus straight b end cell cell space straight b minus straight c end cell cell space straight c minus straight a end cell row cell straight b plus straight c end cell cell space straight c plus straight a end cell cell space space straight a plus straight b end cell end table close vertical bar equals space left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 cell space space space space space 0 end cell 0 row cell straight a minus straight b end cell cell space space space space space space 2 straight b minus straight c minus straight a end cell cell space space space straight b plus straight c minus 2 straight a end cell row cell straight b plus straight c end cell cell space space space straight a minus straight b end cell cell space space straight a minus straight c end cell end table close vertical bar comma space by space straight C subscript 2 space minus space straight C subscript 1 comma space space straight C subscript 3 space minus space straight C subscript 1 equals space left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row cell 2 straight b minus straight c minus straight a end cell cell space space space space space space straight b plus straight c minus 2 straight a end cell row cell straight a minus straight b end cell cell space space space space straight a minus straight c end cell end table close vertical bar equals space left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row cell straight b minus straight c end cell cell space space space straight b minus straight a end cell row cell straight a minus straight b end cell cell space space space straight a minus straight c end cell end table close vertical bar comma space by space straight R subscript 1 plus straight R subscript 2 equals left parenthesis straight a plus straight b plus straight c right parenthesis space open square brackets left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight a minus straight c right parenthesis space minus space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight a right parenthesis close square brackets space equals left parenthesis straight a plus straight b plus straight c right parenthesis space open square brackets straight a space straight b space minus straight b space straight c minus straight c space straight a space plus space straight c squared plus straight a squared plus straight b squared minus 2 ab close square brackets equals space left parenthesis straight a plus straight b plus straight c right parenthesis space left parenthesis straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ca right parenthesis space equals space straight a cubed plus straight b cubed plus straight c cubed minus 3 abc equals space straight R. straight H. straight S.$

For Daily Free Study Material Join wiredfaculty

Determinants

Some More Questions From Determinants Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction