Question

Prove that:
$open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight alpha cell space space straight beta end cell cell space straight gamma end cell row βγ cell space space γα end cell cell space space αβ end cell end table close vertical bar space equals space left parenthesis straight beta minus straight gamma right parenthesis thin space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight alpha minus straight beta right parenthesis$

Solution

Let $increment space equals open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight alpha cell space straight beta end cell cell space space straight gamma end cell row βγ cell space γα end cell cell space space αβ. end cell end table close vertical bar$
   $equals space 1 over αβγ open vertical bar table row straight alpha cell space straight beta end cell cell space space straight gamma end cell row cell straight alpha squared end cell cell space straight beta squared end cell cell space space straight gamma squared end cell row αβγ cell space αβγ end cell cell space αβγ end cell end table close vertical bar$
   $open square brackets Multiplying space straight C subscript 1 comma space straight C subscript 2 comma space straight C subscript 3 comma space by space straight alpha comma space straight beta comma space straight gamma space respectively close square brackets$
$equals space αβγ over αβγ open vertical bar table row straight alpha cell space straight beta end cell cell space space space straight gamma end cell row cell straight alpha squared end cell cell space space straight beta squared end cell cell space space space straight gamma squared end cell row 1 1 cell space space 1 end cell end table close vertical bar comma space$
$taking space straight alpha space straight beta space straight gamma space common space from space third space row.$
$equals left parenthesis negative 1 right parenthesis squared space open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight alpha cell space straight beta end cell cell space space straight gamma end cell row cell straight alpha squared end cell cell space space straight beta squared end cell cell space space straight gamma squared end cell end table close vertical bar$
$open square brackets because space space third space row space passes space over space two space rows close square brackets$
   $equals space open vertical bar table row 1 cell space space space 1 end cell cell space space 1 end cell row straight alpha cell space space straight beta end cell cell space space straight gamma end cell row cell straight alpha squared end cell cell space space straight beta squared end cell cell space space straight gamma squared end cell end table close vertical bar equals space open vertical bar table row 1 0 cell space 0 end cell row straight alpha cell space straight beta minus straight alpha end cell cell space space straight gamma minus straight alpha end cell row cell straight alpha squared end cell cell space space straight beta squared minus straight alpha squared end cell cell space space straight gamma squared minus straight alpha squared end cell end table close vertical bar comma space by space straight C subscript 2 minus straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 equals space open vertical bar table row cell straight beta minus straight alpha end cell cell straight gamma minus straight alpha end cell row cell left parenthesis straight beta minus straight alpha right parenthesis thin space left parenthesis straight beta plus straight alpha right parenthesis end cell cell left parenthesis straight gamma minus straight alpha right parenthesis space left parenthesis straight gamma plus straight alpha right parenthesis end cell end table close vertical bar comma space expanding space by space first space row equals space left parenthesis straight beta minus straight alpha right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis space open vertical bar table row 1 cell space space space space space space space 1 end cell row cell straight beta plus straight alpha end cell cell space space space space space space space straight gamma plus straight alpha end cell end table close vertical bar left square bracket Taking space straight beta minus straight alpha comma space straight gamma minus straight alpha space common space from space first space and space second space column space respectively right square bracket space equals space left parenthesis straight beta minus straight alpha right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight gamma plus straight alpha minus straight beta minus straight alpha right parenthesis space equals space left parenthesis straight beta minus straight alpha right parenthesis thin space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight gamma minus straight beta right parenthesis equals space left parenthesis straight alpha minus straight beta right parenthesis thin space left parenthesis straight beta minus straight gamma right parenthesis thin space left parenthesis straight gamma minus straight alpha right parenthesis.$

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