For what values of k, the system of linear equationsx + y + z =
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    Determinants
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    Using the properties of determinants, solve the following for x:
    open vertical bar table row cell straight x plus 2 end cell cell space space straight x plus 6 end cell cell space straight x minus 1 end cell row cell straight x plus 6 end cell cell straight x minus 1 end cell cell straight x plus 2 end cell row cell straight x minus 1 end cell cell straight x plus 2 end cell cell straight x plus 6 end cell end table close vertical bar space equals space 0

    Let space increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space straight x plus 6 end cell cell space straight x minus 1 end cell row cell straight x plus 6 end cell cell straight x minus 1 end cell cell straight x plus 2 end cell row cell straight x minus 1 end cell cell straight x plus 2 end cell cell straight x plus 6 end cell end table close vertical bar space equals space 0
    Applying space straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 1 space and space straight C subscript 3 space rightwards arrow space straight C subscript 3 space minus straight C subscript 1 increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space 4 end cell cell space space minus 3 end cell row cell straight x plus 6 end cell cell space minus 7 end cell cell space minus 4 end cell row cell straight x minus 1 end cell cell space 3 end cell cell space space space 7 end cell end table close vertical bar Applying space straight R subscript 2 space rightwards arrow straight R subscript 2 minus straight R subscript 1 space and space straight R subscript 3 space rightwards arrow straight R subscript 3 minus straight R subscript 1 space space space increment equals space open vertical bar table row cell straight x plus 2 end cell cell space space 4 end cell cell space space minus 3 end cell row 4 cell space minus 11 end cell cell space minus 1 end cell row cell negative 3 end cell cell space minus 1 end cell cell space space 10 end cell end table close vertical bar space Applying space straight R subscript 2 rightwards arrow straight R subscript 2 space plus space straight R subscript 3 increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space 4 end cell cell space space minus 3 end cell row 1 cell space minus 12 end cell cell space space space 9 end cell row cell negative 3 end cell cell space minus 1 end cell cell space 10 end cell end table close vertical bar Applying space straight R subscript 3 space rightwards arrow straight R subscript 3 plus left parenthesis 3 right parenthesis straight R subscript 2 space increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space 4 end cell cell space minus 3 end cell row 1 cell negative 12 end cell cell space space 9 end cell row 0 cell negative 37 end cell cell space 37 end cell end table close vertical bar Expanding space along space straight C subscript 1

    increment space equals space left parenthesis straight x plus 2 right parenthesis space open vertical bar table row cell negative 12 end cell cell space space 9 end cell row cell negative 37 end cell cell space 37 end cell end table close vertical bar minus 1 open vertical bar table row 4 cell space minus 3 end cell row cell negative 37 end cell cell space space 37 end cell end table close vertical bar increment equals left parenthesis straight x plus 2 right parenthesis thin space left parenthesis negative 444 plus 333 right parenthesis minus 1 left parenthesis 148 minus 111 right parenthesis increment equals space left parenthesis straight x plus 2 right parenthesis thin space left parenthesis negative 111 right parenthesis minus 1 left parenthesis 37 right parenthesis therefore space increment space equals space 0 space equals space minus 111 straight x minus 259 therefore space space straight x space equals space minus 259 over 111 equals negative 7 over 3
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    If open vertical bar table row cell 3 straight x end cell cell space 7 end cell row cell negative 2 end cell cell space 4 end cell end table close vertical bar space equals open vertical bar table row 8 cell space 7 end cell row 6 cell space 4 end cell end table close vertical bar comma find the value of x.

    Given that open vertical bar table row cell 3 straight x end cell cell space space 7 end cell row cell negative 2 end cell cell space 4 end cell end table close vertical bar space equals space open vertical bar table row 8 cell space space space 7 end cell row 6 cell space space space 4 end cell end table close vertical bar
    We need to find the value of x

    open vertical bar table row cell 3 straight x end cell cell space space 7 end cell row cell negative 2 end cell cell space 4 end cell end table close vertical bar space equals space open vertical bar table row 8 cell space space space 7 end cell row 6 cell space space space 4 end cell end table close vertical bar rightwards double arrow space space space 12 straight x minus left parenthesis negative 14 right parenthesis space equals space 32 minus 42 rightwards double arrow space space 12 straight x plus 14 space equals negative 10 rightwards double arrow space space 12 straight x space equals space minus 10 minus 14 rightwards double arrow space 12 straight x space equals space minus 24 rightwards double arrow space space straight x space equals space minus 2

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    Answer

    Use Properties of determinants, prove that:
    open vertical bar table row cell 1 plus straight a end cell cell space 1 end cell cell space 1 end cell row 1 cell 1 plus straight b end cell 1 row 1 1 cell 1 plus straight c end cell end table close vertical bar space equals space abc plus bc plus ca plus ab

    Consider the determinant
    increment space equals open vertical bar table row cell 1 plus straight a end cell 1 1 row 1 cell 1 plus straight b end cell 1 row 1 1 cell 1 plus straight c end cell end table close vertical bar
    Taking abc common outside, we have

    increment equals abc space open vertical bar table row cell 1 over straight a plus 1 end cell cell 1 over straight b end cell cell space 1 over straight c end cell row cell 1 over straight a end cell cell 1 over straight b plus 1 end cell cell 1 over straight c end cell row cell 1 over straight a end cell cell 1 over straight b end cell cell space 1 over straight c plus 1 end cell end table close vertical bar
    Applying the transformation, straight C subscript 1 rightwards arrow straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 comma

    increment equals space abc open vertical bar table row cell 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c space space space end cell cell 1 over straight b end cell cell 1 over straight c end cell row cell 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell cell space 1 over straight b plus 1 end cell cell 1 over straight c end cell row cell 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell cell 1 over straight b end cell cell space space 1 over straight c plus 1 end cell end table close vertical bar

    rightwards double arrow increment equals abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space open vertical bar table row 1 cell 1 over straight b end cell cell 1 over straight c end cell row cell 1 space end cell cell 1 over straight b plus 1 end cell cell 1 over straight c end cell row 1 cell 1 over straight b end cell cell 1 over straight c plus 1 end cell end table close vertical bar
    Apply the transformations, straight R subscript 2 rightwards arrow straight R subscript 1 space and space straight R subscript 3 rightwards arrow straight R subscript 3 minus straight R subscript 1

    increment equals abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space open vertical bar table row 1 cell space 1 over straight b end cell cell 1 over straight c end cell row 1 1 0 row 1 0 1 end table close vertical bar
    Expanding along straight C subscript 1, we have
    increment equals abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses cross times 1 cross times open vertical bar table row 1 0 row 0 1 end table close vertical bar rightwards double arrow space increment equals abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses equals abc plus ab plus bc plus ca

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    Answer
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    For what values of k, the system of linear equations
    x + y + z = 2
    2x + y - z = 3
    3x + 2y + kz = 4
    has a unique solution?

    For space unique space solution space vertical line straight A vertical line space not equal to 0 open vertical bar table row 1 1 1 row 2 1 cell negative 1 end cell row 3 2 straight k end table close vertical bar space not equal to space 0 straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 1 semicolon space straight C subscript 3 rightwards arrow straight C subscript 3 minus straight C subscript 1 open vertical bar table row 1 0 0 row 2 cell negative 1 end cell cell negative 3 end cell row 3 cell negative 1 end cell cell straight k minus 3 end cell end table close vertical bar space not equal to space 0 expansion space along space straight R subscript 1 minus left parenthesis straight k minus 3 right parenthesis minus 3 space not equal to 0 minus straight k plus 3 minus 3 space not equal to 0 straight k not equal to 0
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    Using properties of determinants, prove that
    open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell zx zy row zx cell left parenthesis straight z plus straight y right parenthesis squared end cell xy row zy xy cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar space space equals space 2 xyz left parenthesis straight x plus straight y plus straight z right parenthesis cubed

    open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell zx zy row zx cell left parenthesis straight z plus straight y right parenthesis squared end cell xy row zy xy cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar space equals space 2 xyz left parenthesis straight x plus straight y plus straight z right parenthesis cubed straight L. straight H. straight S comma space Multiplying space straight R subscript 1 comma space straight R subscript 2 space and space straight R subscript 3 space by space straight z comma space straight x space comma space straight y space respectively equals space 1 over xyz space open vertical bar table row cell straight z left parenthesis straight x plus straight y right parenthesis squared end cell zx cell straight z squared straight y end cell row cell zx squared end cell cell straight x left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared straight y end cell row cell zy squared end cell cell xy squared end cell cell straight y left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar taking space common space straight z comma space straight x comma space straight y comma space from space straight C subscript 1 comma end subscript straight C subscript 2 comma end subscript and space straight C subscript 3 equals space xyz over xyz space open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell straight z cell straight z squared end cell row cell straight x squared end cell cell left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared end cell row cell straight y squared end cell cell straight y squared end cell cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar straight C subscript 1 rightwards arrow straight C subscript 2 minus straight C subscript 3 space and space straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 3 taking space common space straight x plus straight y plus straight z space from space straight C subscript 1 space and space straight C subscript 2 left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell straight x plus straight y minus straight z end cell 0 cell straight z squared end cell row 0 cell left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared end cell row cell straight y minus straight z minus straight x end cell cell straight y minus straight z minus straight x end cell cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar straight R subscript 3 rightwards arrow straight R subscript 3 rightwards arrow left parenthesis straight R subscript 1 plus straight R subscript 2 right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell straight x plus straight y minus straight z end cell 0 cell straight z squared end cell row 0 cell straight z plus straight y minus straight x end cell cell straight x squared end cell row cell negative 2 straight x end cell cell negative 2 zx end cell cell 2 xz end cell end table close vertical bar straight C subscript 1 rightwards arrow zC subscript 1 comma space straight C subscript 2 space rightwards arrow xC subscript 3 equals space fraction numerator left parenthesis straight x plus straight y plus straight z right parenthesis squared over denominator xz end fraction space open vertical bar table row cell straight z left parenthesis straight x plus straight y minus straight z right parenthesis end cell 0 cell straight z squared end cell row 0 cell straight x left parenthesis straight z plus straight y minus straight x right parenthesis end cell cell straight x squared end cell row cell negative 2 xz end cell cell negative 2 zx end cell cell 2 xz end cell end table close vertical bar straight C subscript 1 rightwards arrow straight C subscript 1 plus straight C subscript 3 comma space space straight C subscript 2 space rightwards arrow straight C subscript 2 plus straight C subscript 3 equals space fraction numerator left parenthesis straight x plus straight y plus straight z right parenthesis squared over denominator xz end fraction space open vertical bar table row cell straight z left parenthesis straight x plus straight y right parenthesis end cell cell straight z squared end cell cell straight z squared end cell row cell straight x squared end cell cell straight x left parenthesis straight z plus straight y right parenthesis end cell cell straight x squared end cell row 0 0 cell 2 xz end cell end table close vertical bar taking space straight z space and space straight x space common space from space straight R subscript 1 space and space straight R subscript 2 expansion space along space space straight R subscript 3 equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz space left parenthesis left parenthesis straight x plus straight y right parenthesis left parenthesis straight z plus straight y right parenthesis minus xz right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz left parenthesis xz plus xy plus yz plus straight y squared minus xz right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz left parenthesis xy space plus yz space plus straight y squared right parenthesis equals space 2 xyz space left parenthesis straight x plus straight y plus straight z right parenthesis cubed
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