Using properties of determinants, prove that

$open vertical bar table row cell straight a plus straight x end cell straight y cell space space space space straight z end cell row straight x cell space space space straight a plus straight y end cell cell space space space straight z end cell row straight x straight y cell space space straight a plus straight z end cell end table close vertical bar space equals a squared space left parenthesis a plus x plus y plus z right parenthesis$

Solution

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

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