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Determinants

Question
CBSEENMA12034472

If space open vertical bar table row straight a cell space space straight a cubed end cell cell space space straight a to the power of 4 minus 1 end cell row straight b cell space straight b cubed end cell cell space space straight b to the power of 4 minus 1 end cell row straight c cell space straight c cubed end cell cell space space straight c to the power of 4 minus 1 end cell end table close vertical bar space equals space 0 comma
show that
a + b + c = abc(ab + bc + ca) where a , b , c are all different.

Solution
because space space space space open vertical bar table row straight a cell space space straight a cubed end cell cell space space straight a to the power of 4 minus 1 end cell row straight b cell space space straight b cubed end cell cell space space straight b to the power of 4 minus 1 end cell row straight c cell space space straight c cubed end cell cell space space straight c to the power of 4 minus 1 end cell end table close vertical bar space equals space 0
therefore space space space space open vertical bar table row straight a cell space space straight a cubed end cell cell space space straight a to the power of 4 end cell row straight b cell space straight b cubed end cell cell space straight b to the power of 4 end cell row straight c cell space straight c cubed end cell cell space space straight c to the power of 4 end cell end table close vertical bar plus space open vertical bar table row straight a cell space space space space straight a cubed end cell cell space space minus 1 end cell row straight b cell space space space straight b cubed end cell cell space space minus 1 end cell row straight c cell space space space straight c cubed end cell cell space space minus 1 end cell end table close vertical bar space equals space 0
rightwards double arrow space space space space space abc space open vertical bar table row 1 cell space space space straight a squared end cell cell space space space straight a cubed end cell row 1 cell space space straight b squared end cell cell space space straight b cubed end cell row 1 cell space space straight c squared end cell cell space space straight c cubed end cell end table close vertical bar space minus space open vertical bar table row 1 cell space straight a end cell cell space space straight a cubed end cell row 1 cell space straight b end cell cell space straight b cubed end cell row 1 cell space straight c end cell cell space straight c cubed end cell end table close vertical bar space equals space 0
rightwards double arrow space abc space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space left parenthesis ab plus bc plus ca right parenthesis minus left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis. space left parenthesis straight a plus straight b plus straight c right parenthesis space equals space 0
rightwards double arrow space space space space space abc space left parenthesis ab plus bc plus ca right parenthesis space minus space left parenthesis straight a plus straight b plus straight c right parenthesis space equals space 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 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 straight a comma space straight b comma space straight c space are space all space different end cell row cell because space straight a space minus space straight b space not equal to 0 comma space space straight b minus straight c not equal to 0 comma space straight c minus straight a not equal to 0 end cell end table close square brackets
equals space straight a plus straight b plus straight c space equals space abc space left parenthesis ab plus bc plus ca right parenthesis

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