Question

$Evaluate open vertical bar table row straight x straight y cell space space space straight x plus straight y end cell row straight y cell space space space straight x plus straight y end cell straight x row cell straight x plus straight y end cell straight x straight y end table close vertical bar$

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Solution

Let space space increment space equals open vertical bar table row straight x straight y cell space space space straight x plus straight y end cell row straight y cell space space space straight x plus straight y end cell straight x row cell straight x plus straight y end cell straight x straight y end table close vertical bar space equals space open vertical bar table row cell 2 straight x plus 2 straight y end cell cell space space space space space space 2 straight x plus 3 straight y end cell cell space space space space 2 straight x plus 2 straight y end cell row straight y cell straight x plus straight y end cell straight x row cell straight x plus straight y end cell straight x straight y end table close vertical bar space comma space by space straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3 space space space space space space space space space space space space equals space 2 left parenthesis straight x plus straight y right parenthesis space space open vertical bar table row straight x straight y cell space space space straight x plus straight y end cell row straight y cell space space space straight x plus straight y end cell straight x row cell straight x plus straight y end cell straight x straight y end table close vertical bar space space space space space space space space space space space space equals space 2 left parenthesis straight x plus straight y right parenthesis space space open vertical bar table row straight x straight y cell space space space straight x plus straight y end cell row straight y cell space space space straight x plus straight y end cell straight x row cell straight x plus straight y end cell straight x straight y end table close vertical bar space by comma space straight C subscript 2 minus straight C subscript 1 minus straight C subscript 3 minus straight C subscript 1 space space space space space space space space space space space space equals space 2 left parenthesis straight x plus straight y right parenthesis space open vertical bar table row straight x cell space space space straight x minus straight y end cell row cell negative straight y end cell cell negative straight x end cell end table close vertical bar space space space space space space space space space space space space equals space 2 left parenthesis straight x plus straight y right parenthesis space open square brackets table row cell negative straight x squared end cell end table plus xy minus straight y squared close square brackets space space equals space 2 left parenthesis straight x plus straight y right parenthesis space left parenthesis straight x squared minus xy plus straight y squared right parenthesis space equals space 2 left parenthesis straight x cubed plus straight y cubed right parenthesis

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Determinants

$Evaluate open vertical bar table row straight x straight y cell space space space straight x plus straight y end cell row straight y cell space space space straight x plus straight y end cell straight x row cell straight x plus straight y end cell straight x straight y end table close vertical bar$

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