Question

$prove space that space open vertical bar table row cell negative straight a end cell cell space space ab end cell cell space space ac end cell row ba cell space space minus straight b squared end cell cell space space space bc end cell row ca cell space cb end cell cell negative straight c squared end cell end table close vertical bar space equals 4 space straight a to the power of 2 space end exponent straight b squared space straight c squared$

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Solution

Let space increment space space equals space space space open vertical bar table row cell negative straight a end cell cell space space ab end cell cell space space ac end cell row ba cell space space minus straight b squared end cell cell space space space bc end cell row ca cell space cb end cell cell negative straight c squared end cell end table close vertical bar space Taking space straight a comma space straight b comma space straight c space common space from space first comma space second space and space third space column space respectively comma space we space have space space space space space space increment space space space equals space abc space open vertical bar table row cell negative straight a end cell cell space space space space straight a end cell cell space space straight a end cell row straight b cell space minus straight b end cell cell space space straight b end cell row straight c cell space space space space straight c end cell cell space minus straight c end cell end table close vertical bar space space space space space space space space space space space space space equals space straight a squared space straight b squared space straight c space squared space space space space open vertical bar table row cell negative 1 end cell cell space space space space space space 1 end cell cell space space space space space 1 end cell row 1 cell space space minus 1 end cell cell space space space space space 1 end cell row 1 cell space space space space space space 1 end cell cell space minus 1 end cell end table close vertical bar by space taking space straight a. space straight b comma space straight c space common space from space 1 st comma space 2 nd comma space 3 rd space row space respectively. space space space space space space space space space space space space equals straight a squared space straight b squared space straight c squared space space open vertical bar table row cell negative 1 end cell cell space space 0 end cell cell space space space 0 end cell row 1 cell space space 0 end cell cell space space space 2 end cell row 1 cell space space 2 end cell cell space space space 0 end cell end table close vertical bar space comma space by space straight C subscript 2 plus straight C subscript 1 comma space straight C subscript 3 plus straight C subscript 1 space space space space space space space space space space space space equals space straight a squared space straight b squared space straight c squared space space left parenthesis negative 1 right parenthesis space space space open vertical bar table row 0 cell space space space 2 end cell row 2 cell space space space 0 end cell end table close vertical bar space space equals space straight a squared space straight b squared space straight c squared space space left parenthesis negative 1 right parenthesis left parenthesis 0 minus 4 right parenthesis space space space space space space space space space space space space equals space space 4 space straight a squared space straight b squared space straight c squared space

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