Using properties of determinants, prove that
$open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell zx zy row zx cell left parenthesis straight z plus straight y right parenthesis squared end cell xy row zy xy cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar space space equals space 2 xyz left parenthesis straight x plus straight y plus straight z right parenthesis cubed$

open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell zx zy row zx cell left parenthesis straight z plus straight y right parenthesis squared end cell xy row zy xy cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar space equals space 2 xyz left parenthesis straight x plus straight y plus straight z right parenthesis cubed straight L. straight H. straight S comma space Multiplying space straight R subscript 1 comma space straight R subscript 2 space and space straight R subscript 3 space by space straight z comma space straight x space comma space straight y space respectively equals space 1 over xyz space open vertical bar table row cell straight z left parenthesis straight x plus straight y right parenthesis squared end cell zx cell straight z squared straight y end cell row cell zx squared end cell cell straight x left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared straight y end cell row cell zy squared end cell cell xy squared end cell cell straight y left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar taking space common space straight z comma space straight x comma space straight y comma space from space straight C subscript 1 comma end subscript straight C subscript 2 comma end subscript and space straight C subscript 3 equals space xyz over xyz space open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell straight z cell straight z squared end cell row cell straight x squared end cell cell left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared end cell row cell straight y squared end cell cell straight y squared end cell cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar straight C subscript 1 rightwards arrow straight C subscript 2 minus straight C subscript 3 space and space straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 3 taking space common space straight x plus straight y plus straight z space from space straight C subscript 1 space and space straight C subscript 2 left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell straight x plus straight y minus straight z end cell 0 cell straight z squared end cell row 0 cell left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared end cell row cell straight y minus straight z minus straight x end cell cell straight y minus straight z minus straight x end cell cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar straight R subscript 3 rightwards arrow straight R subscript 3 rightwards arrow left parenthesis straight R subscript 1 plus straight R subscript 2 right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell straight x plus straight y minus straight z end cell 0 cell straight z squared end cell row 0 cell straight z plus straight y minus straight x end cell cell straight x squared end cell row cell negative 2 straight x end cell cell negative 2 zx end cell cell 2 xz end cell end table close vertical bar straight C subscript 1 rightwards arrow zC subscript 1 comma space straight C subscript 2 space rightwards arrow xC subscript 3 equals space fraction numerator left parenthesis straight x plus straight y plus straight z right parenthesis squared over denominator xz end fraction space open vertical bar table row cell straight z left parenthesis straight x plus straight y minus straight z right parenthesis end cell 0 cell straight z squared end cell row 0 cell straight x left parenthesis straight z plus straight y minus straight x right parenthesis end cell cell straight x squared end cell row cell negative 2 xz end cell cell negative 2 zx end cell cell 2 xz end cell end table close vertical bar straight C subscript 1 rightwards arrow straight C subscript 1 plus straight C subscript 3 comma space space straight C subscript 2 space rightwards arrow straight C subscript 2 plus straight C subscript 3 equals space fraction numerator left parenthesis straight x plus straight y plus straight z right parenthesis squared over denominator xz end fraction space open vertical bar table row cell straight z left parenthesis straight x plus straight y right parenthesis end cell cell straight z squared end cell cell straight z squared end cell row cell straight x squared end cell cell straight x left parenthesis straight z plus straight y right parenthesis end cell cell straight x squared end cell row 0 0 cell 2 xz end cell end table close vertical bar taking space straight z space and space straight x space common space from space straight R subscript 1 space and space straight R subscript 2 expansion space along space space straight R subscript 3 equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz space left parenthesis left parenthesis straight x plus straight y right parenthesis left parenthesis straight z plus straight y right parenthesis minus xz right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz left parenthesis xz plus xy plus yz plus straight y squared minus xz right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz left parenthesis xy space plus yz space plus straight y squared right parenthesis equals space 2 xyz space left parenthesis straight x plus straight y plus straight z right parenthesis cubed

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