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Determinants

Question
CBSEENMA12034475

Prove that:
open vertical bar table row cell left parenthesis straight b plus straight c right parenthesis squared end cell cell straight a squared end cell cell straight a squared end cell row cell straight b squared end cell cell left parenthesis straight c plus straight a right parenthesis squared end cell cell straight b squared end cell row cell straight c squared end cell cell straight c squared end cell cell left parenthesis straight a plus straight b right parenthesis squared end cell end table close vertical bar space equals space 2 space abc space left parenthesis straight a plus straight b plus straight c right parenthesis cubed.

Solution

 Let space increment space equals space open vertical bar table row cell left parenthesis straight b plus straight c right parenthesis squared end cell cell straight a squared end cell cell straight a squared end cell row cell straight b squared end cell cell left parenthesis straight c plus straight a right parenthesis squared end cell cell straight b squared end cell row cell straight c squared end cell cell straight c squared end cell cell left parenthesis straight a plus straight b right parenthesis squared end cell end table close vertical bar                               ...(1)
Putting a = 0 in (1), we get,
    increment space equals space open vertical bar table row cell left parenthesis straight b plus straight c right parenthesis squared end cell 0 cell space space space 0 end cell row cell straight b squared end cell cell straight c squared end cell cell space space space straight b squared end cell row cell straight c squared end cell cell straight a squared end cell cell space space straight b squared end cell end table close vertical bar space equals space straight b squared straight c squared space open vertical bar table row cell left parenthesis straight b plus straight c right parenthesis squared end cell cell space space space 0 end cell cell space space 0 end cell row cell straight b squared end cell 1 cell space space 1 end cell row cell straight c squared end cell 1 cell space space 1 end cell end table close vertical bar
space space space space space equals space straight b squared straight c squared left parenthesis 0 right parenthesis space equals space 0 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 space space space space space open square brackets because space two space columns space are space identical close square brackets
therefore space space space space space space straight a space minus space 0 space space straight i. straight e. comma space straight a space is space factor space of space increment.
Similarly comma space straight b comma space straight c space are space factors space of space increment.
Again putting a + b + c = 0 in (1), we get,
increment space equals space open vertical bar table row cell left parenthesis negative straight a right parenthesis squared end cell cell space space straight a squared end cell cell space space space space space straight a squared end cell row cell straight b squared end cell cell left parenthesis negative straight b right parenthesis squared end cell cell space space space space straight b squared end cell row cell straight c squared end cell cell straight c squared end cell cell space space left parenthesis negative straight c right parenthesis squared end cell end table close vertical bar space equals space open vertical bar table row cell straight a squared end cell cell space space space straight a squared end cell cell space space straight a squared end cell row cell straight b squared end cell cell space space straight b squared end cell cell space space straight b squared end cell row cell straight c squared end cell cell space space straight c squared end cell cell space space straight c squared end cell end table close vertical bar space equals space 0
therefore space space space left parenthesis straight a plus straight b plus straight c right parenthesis squared space is space straight a space factor space of space increment.
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 space space space space space space space space space space space open square brackets table row cell because space all space the space three space columns space of space increment space become space identical space when space we space put end cell row cell straight a plus straight b plus straight c space equals space 0 end cell end table close square brackets
Now space increment space is space of space sixth space degree
therefore space space space space space increment space has space got space one space more space linear space factor space of space the space type space straight k space left parenthesis straight a plus straight b plus straight c right parenthesis
therefore space space space space space space space space increment space equals space straight k space straight a space straight b space straight c space left parenthesis straight a plus straight b plus straight c right parenthesis cubed 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 space space space space space space space space space... left parenthesis 2 right parenthesis space space space space space space space space space space space space space space
or space space space space space space space space space space space open vertical bar table row cell left parenthesis straight b plus straight c right parenthesis squared end cell cell straight a squared end cell cell straight a squared end cell row cell straight b squared end cell cell left parenthesis straight c plus straight a right parenthesis squared end cell cell straight b squared end cell row cell straight c squared end cell cell straight c squared end cell cell left parenthesis straight a plus straight b right parenthesis squared end cell end table close vertical bar space equals space straight k space straight a space straight b space straight c space left parenthesis straight a plus straight b plus straight c right parenthesis cubed
Put space straight a space equals space straight b space equals space straight c space space equals space 1
therefore space space space space open vertical bar table row 4 cell space space 1 end cell cell space space space 1 end cell row 1 cell space space space 4 end cell cell space space space 1 end cell row 1 cell space space 1 end cell cell space space space 4 end cell end table close vertical bar space equals space 27 space straight k
therefore space space space space open vertical bar table row 0 cell space space 0 end cell cell space space 1 end cell row cell negative 3 end cell cell space 3 end cell cell space space 1 end cell row cell negative 15 end cell cell space space minus 3 end cell cell space space 4 end cell end table close vertical bar space equals space 27 space straight k comma space by space straight C subscript 1 minus 4 space straight C subscript 3 comma space straight C subscript 2 minus straight C subscript 1
therefore space space space space space open vertical bar table row cell negative 3 end cell cell space space space space space space space space 3 end cell row cell negative 15 end cell cell space space space space minus 3 end cell end table close vertical bar space equals space 27 space straight k
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 space space space space space space space space space space space space space space space space


therefore space space 9 plus 45 space equals space 27 space straight k space space space rightwards double arrow space space space 27 space straight k space equals space 54 space space space rightwards double arrow space space space straight k space equals space 2
therefore space space space from space left parenthesis 2 right parenthesis comma space space increment space equals space 2 space abc thin space left parenthesis straight a plus straight b plus straight c right parenthesis cubed
or space space space space space space space space space space space space space space open vertical bar table row cell left parenthesis straight b plus straight c right parenthesis squared end cell cell straight a squared end cell cell straight a squared end cell row cell straight b squared end cell cell left parenthesis straight c plus straight a right parenthesis squared end cell cell straight b squared end cell row cell straight c squared end cell cell straight c squared end cell cell left parenthesis straight a plus straight b right parenthesis squared end cell end table close vertical bar space equals space 2 space straight a space straight b space straight c space left parenthesis straight a plus straight b plus straight c right parenthesis cubed

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