Question

Prove that:
$open vertical bar table row cell straight a squared plus 1 end cell cell straight a space straight b end cell cell straight a space straight c end cell row cell straight a space straight b end cell cell straight b squared plus 1 end cell cell straight b space straight c end cell row cell straight a space straight c end cell cell straight b space straight c end cell cell straight c squared plus 1 end cell end table close vertical bar space equals space 1 plus straight a squared plus straight b squared plus straight c squared.$

Solution

Let space increment space equals space open vertical bar table row cell straight a squared plus 1 end cell cell space straight a space straight b end cell cell straight a space straight c end cell row cell straight a space straight b end cell cell space space straight b squared plus 1 end cell cell straight b space straight c end cell row cell straight a space straight c end cell cell straight b space straight c end cell cell straight c squared plus 1 end cell end table close vertical bar space space space space space space space equals space 1 over abc open vertical bar table row cell straight a left parenthesis straight a squared plus 1 right parenthesis end cell cell ab squared end cell cell straight a space straight c squared end cell row cell straight a squared straight b end cell cell straight b left parenthesis straight b squared plus 1 right parenthesis end cell cell straight b space straight c squared end cell row cell straight a squared straight c end cell cell straight b squared straight c end cell cell straight c left parenthesis straight c squared plus 1 right parenthesis end cell end table close vertical bar comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space by space multiplying space straight C subscript 1 comma space straight C subscript 2 comma space straight C subscript 3 space by space straight a comma space straight b comma space straight c space respectively. space space space space space space equals space abc over abc open vertical bar table row cell straight a squared plus 1 end cell cell space straight b squared end cell cell straight c squared end cell row cell straight a squared end cell cell space straight b squared plus 1 end cell cell straight c squared end cell row cell straight a squared end cell cell straight b squared end cell cell straight c squared plus 1 end cell end table close vertical bar. space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space by space taking space straight a comma space straight b space straight c comma space common space from space straight R subscript 1 comma space straight R subscript 2 comma space straight R subscript 3 space respectively. space space space space equals space open vertical bar table row cell 1 plus straight a squared plus straight b squared plus straight c squared end cell cell space space straight b squared end cell cell space space straight c squared end cell row cell 1 plus straight a squared plus straight b squared plus straight c squared end cell cell space space space space space straight b squared plus 1 end cell cell straight c squared end cell row cell 1 plus straight a squared plus straight b squared plus straight c squared end cell cell straight b squared end cell cell space straight c squared plus 1 end cell end table close vertical bar space comma space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 equals space left parenthesis 1 plus straight a squared plus straight b squared plus straight c squared right parenthesis space open vertical bar table row 1 cell space space straight b squared end cell cell straight c squared end cell row 1 cell space space straight b squared plus 1 end cell cell straight c squared end cell row 1 cell straight b squared end cell cell straight c squared plus 1 end cell end table close vertical bar space space space equals space left parenthesis 1 plus straight a squared plus straight b squared space plus straight c squared right parenthesis thin space space open vertical bar table row 1 cell space space space space straight b squared end cell cell straight c squared end cell row 0 1 0 row 0 0 1 end table close vertical bar comma space space space by space straight R subscript 2 space minus space straight R subscript 1 comma space space space straight R subscript 3 space minus space straight R subscript 1 equals space left parenthesis 1 plus straight a squared plus straight b squared plus straight c squared right parenthesis thin space left square bracket 1.1.1 right square bracket space space space space space space space space space space space space space space space space left square bracket Products space of space diagonal space elements right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 plus straight a squared plus straight b squared plus straight c squared. space space space space space space space space space space space space space space space space

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