Question

Prove that
$open vertical bar table row cell 1 plus straight a end cell 1 1 row 1 cell 1 plus straight b end cell 1 row 1 1 cell 1 plus straight c end cell end table close vertical bar space equals space straight a space straight b space straight c space open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space space or space space straight a space straight b plus straight b space straight c plus straight c space straight a plus straight a space straight b space straight c$

Solution

Let $increment space equals space open vertical bar table row cell 1 plus straight a end cell 1 1 row 1 cell 1 plus straight b end cell 1 row 1 1 cell 1 plus straight c end cell end table close vertical bar$
$equals space abc space open vertical bar table row cell 1 over straight a plus 1 end cell cell 1 over straight a end cell cell 1 over straight a end cell row cell 1 over straight b end cell cell 1 over straight b plus 1 end cell cell 1 over straight b end cell row cell 1 over straight c end cell cell 1 over straight c end cell cell space space 1 over straight c plus 1 end cell end table close vertical bar$
by taking a, b, c common from $straight R subscript 1 comma space straight R subscript 2 comma space straight R subscript 3$ respectively
$equals space abc space open vertical bar table row cell 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell cell space space space space space 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell cell space space space space space 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell row cell 1 over straight b end cell cell 1 over straight b plus 1 end cell cell 1 over straight b end cell row cell 1 over straight c end cell cell 1 over straight c end cell cell space space space space 1 over straight c plus 1 end cell end table close vertical bar$

by $straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3$
$equals space abc space open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space open vertical bar table row 1 1 1 row cell 1 over straight b end cell cell space space space 1 over straight b plus 1 end cell cell 1 over straight b end cell row cell 1 over straight c end cell cell 1 over straight c end cell cell space 1 over straight c plus 1 end cell end table close vertical bar equals space abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space open vertical bar table row 1 cell space space 0 end cell cell space space space 0 end cell row cell 1 over straight b end cell cell space 1 end cell cell space space space 0 end cell row cell 1 over straight c end cell 0 cell space space space 1 end cell end table close vertical bar comma space by space straight C subscript 2 space minus space straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 equals space abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space left square bracket 1. space 1. space 1 space right square bracket space space space space space space space space left square bracket Product space of space diagonal space elements right square bracket equals space abc space open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses.$

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