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Show that:
$open vertical bar table row cell straight b plus straight c end cell cell space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight q plus straight r end cell cell space straight r plus straight p end cell cell space straight p plus straight q end cell row cell straight y plus straight z end cell cell space straight z plus straight x end cell cell space straight x plus straight y end cell end table close vertical bar space equals space 2 open vertical bar table row straight a cell space space space space straight b end cell cell space space straight c end cell row straight p cell space space space straight q end cell cell space space straight r end cell row straight x cell space space straight y end cell cell space space straight z end cell end table close vertical bar$

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102.
Show that:
$open vertical bar table row straight a cell space space straight b end cell cell space space straight c end cell row cell straight a plus 2 straight x end cell cell space straight b plus 2 straight y end cell cell space space space straight c plus 2 space straight z end cell row straight x straight y cell space straight z end cell end table close vertical bar space equals space 0$

straight L. straight H. straight S. space equals space open vertical bar table row straight a cell space space straight b end cell cell space space straight c end cell row cell straight a plus 2 straight x end cell cell space space straight b plus 2 straight y end cell cell space space space straight c plus 2 straight z end cell row straight x straight y straight z end table close vertical bar space space space space space space space space space space space space equals space open vertical bar table row straight a cell space space space straight b end cell cell space space straight c end cell row straight a cell space space straight b end cell cell space space straight c end cell row straight x cell space space straight y end cell cell space space straight z end cell end table close vertical bar space plus space open vertical bar table row straight a straight b straight c row cell 2 space straight x end cell cell 2 space straight y end cell cell 2 space straight z end cell row straight x straight y straight z end table close vertical bar equals space open vertical bar table row straight a cell space space straight b end cell cell space space straight c end cell row straight a cell space space straight b end cell cell space space straight c end cell row straight x cell space space straight y end cell cell space space straight z end cell end table close vertical bar plus space 2 space open vertical bar table row straight a cell space space straight b end cell cell space space straight c end cell row straight x cell space space straight y end cell cell space space straight z end cell row straight x cell space space straight y end cell cell space space straight z end cell end table close vertical bar equals space 0 plus 2 space left parenthesis 0 right parenthesis 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 open square brackets because two space rows space are space identical close square brackets space equals space 0 plus 0 space equals space 0 space equals space straight R. straight H. straight S.

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103. Factorise the determinant:

open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight a cell space space straight b end cell cell space straight c end cell row cell straight a squared end cell cell space space straight b squared end cell cell space space straight c squared end cell end table close vertical bar.

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Long Answer Type

104.

Prove that:
$open vertical bar table row 1 1 1 row cell straight a squared end cell cell straight b squared end cell cell straight c squared end cell row cell straight a cubed end cell cell straight b cubed end cell cell straight c cubed end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a space straight b space plus space straight b space straight c space plus space straight c space straight a right parenthesis.$

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Short Answer Type

105.

By using properties of determinants, show that:
$open vertical bar table row 1 cell space space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space straight b end cell cell space space space straight b squared end cell row 1 cell space space straight c end cell cell space space space straight c squared end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis space left parenthesis straight c minus straight a right parenthesis$

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106.

Prove the following identities:
$open vertical bar table row 1 cell space space space 1 end cell cell space space space 1 end cell row straight a cell space space straight b end cell cell space space space straight c end cell row cell straight a cubed end cell cell space space space straight b cubed end cell cell space space space straight c cubed end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a plus straight b plus straight c right parenthesis$

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107.

Prove the following identities:
$open vertical bar table row 1 cell space space space straight a end cell cell space space space straight a cubed end cell row 1 cell space space straight b end cell cell space space straight b cubed end cell row 1 cell space space straight c end cell cell space space straight c cubed end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a plus straight b plus straight c right parenthesis$

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108.

Prove the following identities:
$open vertical bar table row 1 cell space space space straight x end cell cell space space space straight x cubed end cell row 1 cell space space straight y end cell cell space space straight y cubed end cell row 1 cell space straight z end cell cell space space straight z cubed end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight x plus straight y plus straight z right parenthesis$

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109.

Using properties of determinants, prove that:
$open vertical bar table row straight x cell space space space straight y end cell cell space space space straight z end cell row cell straight x squared end cell cell space space straight y squared end cell cell space space space straight z squared end cell row cell straight x cubed end cell cell space space straight y cubed end cell cell space space straight z cubed end cell end table close vertical bar space equals space space straight x space straight y space straight z space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis$

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Long Answer Type

110.

Prove that:
$open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight alpha cell space space straight beta end cell cell space straight gamma end cell row βγ cell space space γα end cell cell space space αβ end cell end table close vertical bar space equals space left parenthesis straight beta minus straight gamma right parenthesis thin space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight alpha minus straight beta right parenthesis$

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