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Determinants

Short Answer Type

81.

show that
$open vertical bar table row cell straight x minus straight y minus straight z end cell cell 2 straight x end cell cell 2 straight x end cell row cell 2 straight y end cell cell space space space straight y minus straight z minus straight x end cell cell 3 straight x end cell row cell 2 straight z end cell cell 2 space straight z end cell cell space space straight z minus straight x minus straight y end cell end table close vertical bar space equals space left parenthesis x plus y plus z right parenthesis cubed$

72 Views

82.

show space that open vertical bar table row cell straight a plus straight b plus nc end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis space straight c end cell cell straight b plus straight c plus na end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis straight c end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell straight c plus straight a plus nb end cell end table close vertical bar space equals straight n left parenthesis straight a plus straight b plus straight c right parenthesis cubed

68 Views

83.

show space that open vertical bar table row 1 cell space space space space straight b plus straight c space end cell cell space space space space space straight b squared plus straight c squared end cell row 1 cell space space straight c plus straight a end cell cell space space space straight c squared plus straight a squared end cell row 1 cell space space straight a plus straight b end cell cell space space space space straight a squared plus straight b squared end cell end table close vertical bar

69 Views

84.

prove space that open vertical bar table row cell 1 plus straight a squared plus straight b squared end cell cell 2 ab end cell cell negative 2 straight b end cell row cell 2 ab end cell cell 1 minus straight a squared space plus straight b squared end cell cell 2 straight a end cell row cell 2 straight b end cell cell negative 2 straight a end cell cell 1 minus straight a squared minus straight b squared end cell end table close vertical bar space equals left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis cubed

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85.

By space using space properties space of space determinants comma space show space that open vertical bar table row cell straight x plus 4 end cell cell 2 space straight x end cell cell 2 space straight x end cell row cell 2 space straight x end cell cell space space straight x plus 4 end cell cell 2 space straight x end cell row cell 2 space straight x end cell cell 2 space straight x end cell cell space space straight x plus 4 end cell end table close vertical bar space equals space left parenthesis 5 space straight x plus 4 right parenthesis space left parenthesis 4 minus straight x right parenthesis squared

66 Views

86.

space By space using space properties space of space determinants comma space show space that space space space space space space space space open vertical bar table row cell straight y plus straight k end cell straight y straight y row straight y cell space space space straight y plus straight k end cell straight y row straight y straight y cell space space straight y plus straight k end cell end table close vertical bar space equals straight k squared space left parenthesis 3 straight y plus straight k right parenthesis

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87. Using the properties of determinants, prove that

open vertical bar table row cell straight a plus straight b plus straight c end cell cell negative straight c end cell cell negative straight b end cell row cell negative straight c end cell cell space space space straight a plus straight b plus straight c end cell cell negative straight a end cell row cell negative straight b end cell cell negative straight a end cell cell space space straight a plus straight b plus straight c end cell end table close vertical bar space equals 2 left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis

73 Views

Long Answer Type

88.

show space that space open vertical bar table row cell negative straight a left parenthesis straight b squared plus straight c squared minus straight a squared right parenthesis end cell cell 2 straight b cubed end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell space space space minus straight b left parenthesis straight c squared plus straight a squared plus straight b squared right parenthesis end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell 2 straight b cubed end cell cell space space space minus straight c left parenthesis straight a squared plus straight b squared space minus straight c squared right parenthesis end cell end table close vertical bar space equals abc space left parenthesis straight a squared plus straight b squared plus straight c squared right parenthesis

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89. $prove space that space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space equals left parenthesis straight b squared minus ac right parenthesis left parenthesis ax squared plus 2 space straight b space straight x space straight y space plus straight c space straight y cubed right parenthesis$

Let space space increment space equals space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space space space space space space space space space space space space space equals space begin inline style 1 over xy end style space space space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space space Multiplying space straight C subscript 1 space end subscript by space straight x. straight C subscript 2 space by space straight y space space space space space space space space space space space space equals space begin inline style 1 over xy end style space space space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space by space straight C subscript 1 space end subscript plus. straight C subscript 2 minus straight C subscript 3 space space space space space space space space space space space space equals space begin inline style 1 over xy space space left parenthesis straight a space straight x end style squared space plus 2 space straight b space straight x space straight y space plus straight c space straight y squared right parenthesis space open vertical bar table row cell straight b space straight y end cell cell straight a space straight x plus straight b space straight y end cell row cell straight c space straight y end cell cell straight b space straight x plus straight c space straight y end cell end table close vertical bar space comma space expanding space by space first space column space space space space space space space space space space space space equals space begin inline style fraction numerator straight a space straight x squared plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction end style open square brackets table row cell by space left parenthesis straight b space straight x plus straight c space straight y right parenthesis minus cy left parenthesis straight a space straight x plus straight b space straight y right parenthesis end cell end table close square brackets space space space space space space space space space space space space equals space begin inline style fraction numerator straight a space straight x squared plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction space open square brackets table row cell straight b squared end cell end table space straight x space straight y space plus straight b space straight c space straight y squared space minus straight c space straight a space straight x space straight y space minus straight b space straight c space straight y squared close square brackets end style space space space space space space space space space space space space equals space begin inline style fraction numerator straight a space straight x squared plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction space space open square brackets table row cell straight b squared end cell end table xy minus space straight c space straight a space straight x space straight y close square brackets space equals space fraction numerator straight a space straight x squared space plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction space space open square brackets table row cell left parenthesis straight b squared minus ca right parenthesis space straight x space straight y end cell end table close square brackets space end style space space space space space space space space space space space space equals space space space left parenthesis straight b squared space minus ca right parenthesis space right parenthesis left parenthesis straight a space straight x squared space plus 2 space straight b space straight x space straight y plus straight c space straight y squared space right parenthesis space space space space space space space space space space space space space space space space space

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Short Answer Type

90.

prove space that space space increment space equals space open vertical bar table row cell straight a plus straight b space straight x end cell cell straight c plus straight d space straight x end cell cell space space space straight p plus straight q space straight x end cell row cell straight a space straight x plus straight b end cell cell straight c space straight x plus straight d end cell cell space space straight p space straight x plus straight q end cell row straight upsilon straight nu straight omega end table close vertical bar space equals space left parenthesis 1 minus straight x squared right parenthesis space space open vertical bar table row straight a cell space straight c end cell cell space space straight p end cell row straight b cell space space space straight d space space end cell cell space straight q end cell row straight u cell space straight v end cell cell space space straight w end cell end table close vertical bar

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