Question

$Using space properties space of space determinants comma space prove space that space open vertical bar table row cell straight y plus straight z end cell cell space space space straight x end cell cell space space straight y end cell row cell straight z plus straight x end cell cell space space straight z end cell cell space space straight x end cell row cell straight x plus straight y end cell cell space space straight y end cell cell space straight z end cell end table close vertical bar space equals space left parenthesis straight x plus straight y plus straight z right parenthesis space left parenthesis straight x minus straight z right parenthesis squared$

Solution

straight L. straight H. straight S space space space equals space open vertical bar table row cell straight y plus straight z end cell cell space space space straight x end cell cell space space straight y end cell row cell straight z plus straight x end cell cell space space straight z end cell cell space space straight x end cell row cell straight x plus straight y end cell cell space space straight y end cell cell space straight z end cell end table close vertical bar space equals space open vertical bar table row cell 2 left parenthesis straight x plus straight y plus straight z right parenthesis end cell cell space space space straight x plus straight y plus straight z end cell cell space space straight x plus straight y plus straight z end cell row cell straight z plus straight x end cell straight z straight x row cell straight x plus straight y end cell straight y straight z end table close vertical bar space comma space by space straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3 space space space space space space space space space space space space space space equals space left parenthesis straight x plus straight y plus straight z right parenthesis space space open vertical bar table row 2 cell space 1 end cell cell space space 1 end cell row cell straight z plus straight x end cell cell space space straight z end cell cell space space straight x end cell row cell straight x plus straight y end cell cell space space straight y end cell cell space space straight z end cell end table close vertical bar space space space space space space space space space space space space space space space equals space left parenthesis straight x plus straight y plus straight z right parenthesis space open vertical bar table row 0 1 cell space 1 end cell row cell straight z minus straight x end cell cell straight z minus straight x end cell cell space space straight x end cell row cell straight x plus straight y minus 2 straight z end cell cell straight y minus straight z end cell cell space space straight z end cell end table close vertical bar space comma space by space straight C subscript 1 minus 2 straight C subscript 3 space comma space straight C subscript 2 minus straight C subscript 3 space space space space space space space space space space space space space space equals space left parenthesis straight x plus straight y plus straight z right parenthesis space space open vertical bar table row cell straight z minus straight x end cell cell space space straight z minus straight x end cell row cell straight x plus straight y minus 2 straight z end cell cell space space straight y minus straight z end cell end table close vertical bar space space space space space space space space space space space space space space space equals space left parenthesis straight z minus straight x right parenthesis space left parenthesis straight x plus straight y plus straight z right parenthesis space space space open vertical bar table row 1 1 row cell straight x plus straight y minus 2 straight z end cell cell space space straight y minus straight z end cell end table close vertical bar space space space space space space space space space space space space space space equals space space left parenthesis straight z minus straight x right parenthesis space left parenthesis straight x plus straight y plus straight z right parenthesis space open square brackets table row cell straight y minus straight z minus straight x minus straight y plus 2 straight z end cell end table close square brackets space equals space left parenthesis straight z minus straight x right parenthesis left parenthesis straight x plus straight y plus straight z right parenthesis left parenthesis straight z minus straight x right parenthesis space space space space space space space space space space space space space equals space straight R. straight H. straight S

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