Question

$If space straight a comma space straight b comma space straight c space are space positive space and space unequal comma space show space that space value space of space the space determinant increment space space equals space open vertical bar table row straight a cell space space space straight b end cell cell space space space space straight c end cell row straight b cell space space space straight c end cell cell space space space space straight a end cell row straight c cell space space space straight a end cell cell space space space space straight b end cell end table close vertical bar space is space negative.$

Solution

Let space increment space equals space open vertical bar table row straight a cell space space straight b end cell cell space space straight c end cell row straight b cell space space straight c end cell cell space space straight a end cell row straight c cell space space straight a end cell cell space space straight b end cell end table close vertical bar space equals straight a open vertical bar table row straight c cell space space straight a end cell row straight a cell space space space straight b end cell end table close vertical bar space minus straight b open vertical bar table row straight b cell space space space straight a end cell row straight c cell space space space straight b end cell end table close vertical bar space plus straight c space open vertical bar table row straight b cell space space space straight c end cell row straight c cell space space space straight a end cell end table close vertical bar space space space space space space space space space space equals space space straight a left parenthesis bc minus straight a squared right parenthesis space minus straight b space left parenthesis straight b squared minus ca right parenthesis plus straight c left parenthesis ab minus straight c squared right parenthesis space space space space space space space space space space equals space space abc minus straight a cubed minus straight b cubed space plus abc plus abc minus straight c cubed space equals 3 abc minus straight a cubed minus straight b cubed space minus straight c cubed space space space space space space space space space space equals space minus space open square brackets table row cell straight a cubed end cell end table plus straight b cubed plus straight c cubed minus 3 abc close square brackets space space space space space space space space space space equals space minus space 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 space space space space space equals space minus 1 half space left parenthesis straight a plus straight b plus straight c right parenthesis open square brackets table row cell 2 straight a squared plus 2 straight b squared end cell end table plus 2 straight c squared minus 2 ab minus 2 bc minus 2 ca close square brackets space space space space space space space space space space equals space minus 1 half space left parenthesis straight a plus straight b plus straight c right parenthesis open square brackets table row cell left parenthesis straight a squared plus straight b squared minus 2 ab right parenthesis plus left parenthesis straight b squared end cell end table plus straight c squared minus 2 bc right parenthesis plus left parenthesis straight c squared plus straight a squared minus 2 ca right parenthesis close square brackets space space space space space space space space space equals space minus 1 half space left parenthesis straight a plus straight b plus straight c right parenthesis space open square brackets table row cell left parenthesis straight a minus straight b right parenthesis squared end cell end table plus left parenthesis straight b minus straight c right parenthesis squared plus straight c left parenthesis straight c minus straight a squared right parenthesis close square brackets space less than 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 space open square brackets table row cell therefore space straight a comma space straight b comma space straight c space are space all space positive space and space not space equal end cell end table close square brackets

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