Question

Prove that

$open square brackets table row cell straight x plus straight a end cell straight b straight c row straight a cell space space space straight x plus straight b end cell straight c row straight a straight b cell space space space straight x plus straight c end cell end table close square brackets space equals space x squared space left parenthesis x plus a plus b plus c right parenthesis$

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Solution

straight L. straight H. straight S equals open square brackets table row cell straight x plus straight a end cell straight b straight c row straight a cell space space space straight x plus straight b end cell straight c row straight a straight b cell space space space straight x plus straight c end cell end table close square brackets space equals space open vertical bar table row cell straight x plus straight a plus straight b plus straight c end cell straight b straight c row cell straight x plus straight a plus straight b plus straight c end cell cell space space straight x plus straight b end cell straight c row cell straight x plus straight a plus straight b plus straight c end cell straight b cell space space straight x plus straight c end cell end table close vertical bar space comma space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 space space space space space space space space space space space equals left parenthesis straight x plus straight a plus straight b plus straight c right parenthesis space space open vertical bar table row 1 straight b straight c row 1 cell space space space space straight x plus straight b end cell straight c row 1 straight b cell space space space space straight x plus straight c end cell end table close vertical bar space space space space space space space space space space space equals left parenthesis straight x plus straight a plus straight b plus straight c right parenthesis space space open vertical bar table row 1 cell space space space space space straight b end cell cell space space space space straight c end cell row 0 cell space space space space straight x end cell cell space space space space 0 end cell row 0 cell space space space space 0 end cell cell space space space space straight x end cell end table close vertical bar space comma space by space straight R subscript 2 minus straight R subscript 1 comma space straight R subscript 3 minus straight R subscript 1 space space space space space space space space space space space space equals left parenthesis straight x plus straight a plus straight b plus straight c right parenthesis space space open square brackets table row cell left parenthesis 1 right parenthesis left parenthesis straight x right parenthesis left parenthesis straight x right parenthesis end cell end table close square brackets space space space space space space space space space space space space open square brackets table row cell product space of space diagonal space elements end cell end table close square brackets space space space space space space space space space space space space space equals space straight x squared space left parenthesis straight x plus straight a plus straight b plus straight c right parenthesis space space space space space space space space space space space space equals space straight R. straight H. straight S space space space space space space space space space space space space space space space space space space

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