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Determinants

Question
CBSEENMA12034408

Show that:
increment space equals space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x space straight y end cell cell straight z space straight x end cell row cell straight x space straight y end cell cell left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y space straight z end cell row cell straight x space straight z end cell cell straight y space straight z end cell cell left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar space equals space 2 xyz space left parenthesis straight x plus straight y plus straight z right parenthesis squared

Solution
increment space equals space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x space straight y end cell cell straight z space straight x end cell row cell straight x space straight y end cell cell left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y space straight z end cell row cell straight x space straight z end cell cell straight y space straight z end cell cell left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar
Applying straight R subscript 1 space rightwards arrow space xR subscript 1 comma space straight R subscript 2 rightwards arrow straight y space straight R subscript 2 comma space straight R subscript 3 space rightwards arrow space zR subscript 3 to increment and dividing by x y z , we get
                           increment space equals space 1 over xyz open vertical bar table row cell straight x left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x squared straight y end cell cell straight x squared straight z end cell row cell straight x space straight y end cell cell straight y left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y squared straight z end cell row cell straight x space straight z squared end cell cell straight y space straight z squared end cell cell straight z left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar
Taking common factors, x y, z, from straight C subscript 1 comma space straight C subscript 2 space and space straight C subscript 3 comma space respectively, we get
increment space equals space xyz over xyz space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x squared end cell cell straight x squared end cell row cell straight y squared end cell cell left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y squared end cell row cell straight z squared end cell cell straight z squared end cell cell left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar
Again straight C subscript 2 space rightwards arrow space straight C subscript 2 space minus space straight C subscript 1 comma space space straight C subscript 3 space minus space straight C subscript 1 comma we get
                         increment space equals space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared space space space space end cell cell straight x squared minus left parenthesis straight y plus straight z right parenthesis squared end cell cell space space space space space straight x squared minus left parenthesis straight y plus straight z right parenthesis squared end cell row cell straight y squared end cell cell left parenthesis straight x plus straight z right parenthesis squared minus straight y squared end cell 0 row cell straight z squared end cell 0 cell left parenthesis straight x plus straight y right parenthesis squared minus straight z squared end cell end table close vertical bar
Taking common factor (x + y + z) from straight C subscript 2 space and space straight C subscript 3, we have
                      increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell space space space straight x minus left parenthesis straight y plus straight z right parenthesis end cell cell space space space space straight x minus left parenthesis straight y plus straight z right parenthesis end cell row cell straight y squared end cell cell space left parenthesis straight x plus straight z right parenthesis minus straight y end cell cell space space 0 end cell row cell straight z squared end cell cell space 0 end cell cell space space space space left parenthesis straight x plus straight y right parenthesis minus straight z end cell end table close vertical bar
Applying straight R subscript 1 space rightwards arrow space straight R subscript 1 space minus space left parenthesis straight R subscript 2 space plus space straight R subscript 3 right parenthesis comma space we space have
                   increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell 2 space straight y space straight z end cell cell negative 2 space straight z end cell cell negative 2 straight y end cell row cell straight y squared end cell cell straight x minus straight y plus straight z end cell 0 row cell straight z squared end cell 0 cell straight x plus straight y minus straight z end cell end table close vertical bar
Applying straight C subscript 2 space rightwards arrow space open parentheses straight C subscript 2 plus 1 over straight y straight C subscript 1 close parentheses space and space straight C subscript 3 space rightwards arrow space open parentheses straight C subscript 3 plus 1 over straight z straight C subscript 1 close parentheses comma we get
                   increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell 2 yz end cell 0 0 row cell straight y squared end cell cell straight x plus straight z end cell cell straight y squared over straight z end cell row cell straight z squared end cell cell straight z squared over straight y end cell cell straight x plus straight y end cell end table close vertical bar
increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space left parenthesis 2 yz right parenthesis thin space left square bracket left parenthesis straight x plus straight z right parenthesis thin space left parenthesis straight x plus straight y right parenthesis space minus space yz right square bracket
                                                                               [Expanding by first row]
       equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space left parenthesis 2 space space straight y space straight z right parenthesis thin space left parenthesis straight x squared plus straight x space straight y plus straight x space straight z right parenthesis
equals space left parenthesis straight x plus straight y plus straight z right parenthesis cubed space left parenthesis 2 space straight x space straight y space straight z right parenthesis
equals space 2 space straight x space straight y space straight z space left parenthesis straight x plus straight y plus straight z right parenthesis cubed

                      

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