Show that:
$open vertical bar table row 1 cell space space straight x end cell cell space space yz end cell row 1 cell space straight y end cell cell space space zx end cell row 1 cell space straight z end cell cell space space xy end cell end table close vertical bar space equals space open vertical bar table row 1 cell space space space straight x end cell cell space space straight x squared end cell row 1 cell space space straight y end cell cell space space straight y squared end cell row 1 cell space straight z end cell cell space space straight z squared end cell end table close vertical bar$
and hence factorise.

Solution

Let $Let space increment space equals space open vertical bar table row 1 cell space space straight x end cell cell space space yz end cell row 1 cell space space straight y end cell cell space space zx end cell row 1 cell space straight z end cell cell space space xy end cell end table close vertical bar space equals space 1 over xyz open vertical bar table row straight x cell space space straight x squared end cell cell space xyz end cell row cell straight y space end cell cell straight y squared end cell cell space xyz end cell row straight z cell straight z squared end cell cell space xyz end cell end table close vertical bar comma$
by multiplying $straight R subscript 1 semicolon space straight R subscript 2 comma space straight R subscript 3 space with space straight a comma space straight b comma space straight c space respectively$
$equals space xyz over xyz open vertical bar table row straight x cell space space space straight x squared end cell cell space space 1 end cell row straight y cell space space straight y squared end cell cell space space 1 end cell row straight z cell space space straight z squared end cell cell space space 1 end cell end table close vertical bar$
$equals space open vertical bar table row 1 cell space space straight x end cell cell space space straight x squared end cell row 1 cell space space straight y end cell cell space space straight y squared end cell row 1 straight z cell space space straight z squared end cell end table close vertical bar comma space space by space passing space straight C subscript 3 space over space two space columns.$
Proceed as in (a) (i) by changing a to x, b to y, c to z.

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