Question

Using properties of determinants prove the following:
$open vertical bar table row 1 cell space space straight x end cell cell space space straight x squared end cell row cell straight x squared end cell cell space 1 end cell straight x row straight x cell space straight x end cell 1 end table close vertical bar space equals space left parenthesis 1 minus straight x cubed right parenthesis squared$

Solution

increment space equals space open vertical bar table row 1 cell space space straight x end cell cell space straight x squared end cell row cell straight x squared end cell cell space 1 end cell straight x row straight x cell space straight x squared end cell 1 end table close vertical bar

Applying

straight R subscript 1 rightwards arrow straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3 comma space

we have

increment space equals space open vertical bar table row cell 1 plus straight x plus straight x squared end cell cell 1 plus straight x plus straight x squared end cell cell 1 plus straight x plus straight x squared end cell row cell straight x squared end cell 1 straight x row straight x cell straight x to the power of 21 end cell blank end table close vertical bar equals space left parenthesis 1 plus straight x plus straight x squared right parenthesis space open vertical bar table row 1 cell space space 1 end cell cell space space space 1 end cell row cell straight x squared end cell cell space space 1 end cell cell space space straight x end cell row straight x cell space space straight x squared end cell cell space space space 1 end cell end table close vertical bar

Applying space straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 1 space and space straight C subscript 3 minus straight C subscript 1 comma space we space have colon increment space equals space left parenthesis 1 plus straight x plus straight x squared right parenthesis space open vertical bar table row 1 cell space space space 0 end cell cell space space 0 end cell row cell straight x squared end cell cell space 1 minus straight x squared end cell cell space space straight x minus straight x squared end cell row straight x cell space 1 plus straight x squared end cell cell space 1 minus straight x end cell end table close vertical bar equals left parenthesis 1 plus straight x plus straight x squared right parenthesis thin space left parenthesis 1 minus straight x right parenthesis left parenthesis 1 minus straight x right parenthesis space open vertical bar table row 1 cell space 0 end cell cell space 0 end cell row cell straight x squared end cell cell 1 plus straight x end cell cell space straight x end cell row straight x cell negative straight x end cell cell space 1 end cell end table close vertical bar equals left parenthesis 1 minus straight x cubed right parenthesis left parenthesis 1 minus straight x right parenthesis open vertical bar table row 1 cell space 0 end cell cell space 0 end cell row cell straight x squared space end cell cell 1 plus straight x end cell cell space straight x end cell row straight x cell negative straight x end cell cell space 1 end cell end table close vertical bar

Expanding along R_1,we have:_{$increment equals left parenthesis 1 minus straight x cubed right parenthesis left parenthesis 1 minus straight x right parenthesis thin space left parenthesis 1 right parenthesis space open vertical bar table row cell 1 plus straight x end cell cell space space space straight x end cell row cell negative straight x end cell cell space space 1 end cell end table close vertical bar space space space equals left parenthesis 1 minus straight x cubed right parenthesis left parenthesis 1 minus straight x right parenthesis left parenthesis 1 plus straight x plus straight x squared right parenthesis space space space equals left parenthesis 1 minus straight x cubed right parenthesis left parenthesis 1 minus straight x cubed right parenthesis space space space equals left parenthesis 1 minus straight x cubed right parenthesis squared Hence space proved.$}

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