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?

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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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