Question

Prove that:
$open vertical bar table row cell negative 2 straight a end cell cell space space space space straight a plus straight b end cell cell space space space straight a plus straight c end cell row cell straight b plus straight a end cell cell space space minus 2 straight b end cell cell space space space straight b plus straight c end cell row cell straight c plus straight a end cell cell space space straight c plus straight b end cell cell space space minus 2 straight c end cell end table close vertical bar space equals space 4 left parenthesis straight a plus straight b right parenthesis thin space left parenthesis straight b plus straight c right parenthesis thin space left parenthesis straight c plus straight a right parenthesis.$

Solution

Let space increment space equals space open vertical bar table row cell negative 2 straight a end cell cell space space space space space space straight a plus straight b end cell cell space space space space straight a plus straight c end cell row cell straight b plus straight a end cell cell space space minus 2 straight b end cell cell space space space straight b plus straight c end cell row cell straight c plus straight a end cell cell space straight c plus straight b end cell cell space space space minus 2 straight c end cell end table close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Putting space straight a plus straight b space equals space 0 space space space or space space straight a space equals negative straight b space in space left parenthesis 1 right parenthesis comma space we space get comma increment space equals space open vertical bar table row cell 2 straight b end cell cell space space 0 end cell cell space space space space straight c minus straight b end cell row 0 cell space space minus 2 straight b end cell cell space space space space straight b plus straight c end cell row cell straight c minus straight b end cell cell space space space straight c plus straight b end cell cell space space space minus 2 straight c end cell end table close vertical bar space space space space equals space 2 straight b open vertical bar table row cell negative 2 straight b end cell cell space space space space space space space space space space straight b plus straight c end cell row cell straight c plus straight b end cell cell space space space space space space minus 2 straight c end cell end table close vertical bar plus left parenthesis straight c minus straight b right parenthesis space open vertical bar table row 0 cell space space space space space space space minus 2 straight b end cell row cell straight c minus straight b end cell cell space space space space space space space space space straight c plus straight b end cell end table close vertical bar space space space space space equals space 2 space straight b open square brackets 4 space straight b space straight c space minus straight b squared minus straight c squared minus 2 space straight b space straight c close square brackets space plus space left parenthesis straight c minus straight b right parenthesis space left square bracket 0 plus 2 space straight b space straight c minus 2 space straight b squared right square bracket space space space space equals 2 space straight b space left parenthesis negative straight b squared minus straight c squared plus 2 bc right parenthesis plus left parenthesis straight c minus straight b right parenthesis thin space left parenthesis 2 space straight b space straight c space minus space 2 space straight b squared right square bracket space space space space equals space minus 2 straight b cubed minus 2 bc squared plus 4 straight b squared straight c minus 2 bc squared minus 2 straight b squared straight c minus 2 straight b squared straight c plus 2 straight b cubed space space space space space equals space 0 therefore space space space space space space space straight a plus straight b minus 0 space space space space space straight i. straight e. space space space straight a plus straight b space is space straight a space factor space of space increment Similarly space straight b plus straight c comma space space space straight c plus straight a space space space space are space factors space of space increment. Now space increment space is space of space 3 rd space degree

therefore space space space space space space increment space equals space straight k space left parenthesis straight a plus straight b right parenthesis thin space left parenthesis straight b plus straight c right parenthesis thin space left parenthesis straight c plus straight a right parenthesis

...(2)

or space space open vertical bar table row cell negative 2 straight a end cell cell space space space straight a plus straight b end cell cell space space space space space straight a plus straight c end cell row cell straight b plus straight a end cell cell space space space minus 2 straight b end cell cell space space space space straight b plus straight c end cell row cell straight c plus straight a end cell cell space space straight c plus straight b end cell cell space space space minus 2 straight c end cell end table close vertical bar space equals space straight k left parenthesis straight a plus straight b right parenthesis thin space left parenthesis straight b plus straight c right parenthesis thin space left parenthesis straight c plus straight a right parenthesis Put space straight a space equals space 0 comma space space space straight b space equals space 1 comma space space straight c space equals space 1 therefore space space space space open vertical bar table row 0 cell space space space 1 end cell cell space space space 1 end cell row 1 cell space space minus 2 end cell cell space space space space space 2 end cell row 1 cell space space 2 end cell cell space space minus 2 end cell end table close vertical bar space equals space straight k left parenthesis 0 plus 1 right parenthesis thin space left parenthesis 1 plus 1 right parenthesis thin space left parenthesis 1 plus 0 right parenthesis therefore space space space space open vertical bar table row 0 cell space space space space space space space 0 end cell cell space space space space space space 1 end cell row 1 cell space space space minus 4 end cell cell space space space space space space 2 end cell row 1 cell space space space space space space 4 end cell cell space space space minus 2 end cell end table close vertical bar space equals space 2 space straight k space space space space space space space rightwards double arrow space space space space space space open vertical bar table row 1 cell space space space space minus 4 end cell row 1 cell space space space space space space space 4 end cell end table close vertical bar space equals space 2 space straight k rightwards double arrow space space space 4 plus 4 space equals space 2 space straight k space space space space space space space space rightwards double arrow space space space space space space space 8 space equals space 2 space straight k space space space space space space space rightwards double arrow space space space straight k space equals space 4 therefore space space space from space left parenthesis 2 right parenthesis comma space space space increment space equals space 4 left parenthesis straight a plus straight b right parenthesis thin space left parenthesis straight b plus straight c right parenthesis thin space left parenthesis straight c plus straight a right parenthesis or space space space space space space open vertical bar table row cell negative 2 straight a end cell cell space space space straight a plus straight b end cell cell space space space space straight a plus straight c end cell row cell straight b plus straight a end cell cell space space minus 2 straight b end cell cell space space space straight b plus straight c end cell row cell straight c plus straight a end cell cell space space straight c plus straight b end cell cell space space minus 2 straight c end cell end table close vertical bar space equals space 4 left parenthesis straight a plus straight b right parenthesis thin space left parenthesis straight b plus straight c right parenthesis thin space left parenthesis straight c plus straight a right parenthesis.

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