Using properties of determinants, prove that

$open vertical bar table row cell straight x space end cell cell straight x plus straight y space end cell cell straight x plus 2 straight y space end cell row cell straight x plus 2 straight y space end cell cell straight x space end cell cell straight x plus straight y end cell row cell straight x plus straight y space end cell cell straight x plus 2 straight y space end cell cell straight x space end cell end table close vertical bar space equals space 9 straight y squared left parenthesis straight x plus straight y right parenthesis$

Solution

open vertical bar table row cell straight x space end cell cell straight x plus straight y space end cell cell straight x plus 2 straight y space end cell row cell straight x plus 2 straight y space end cell cell straight x space end cell cell straight x plus straight y end cell row cell straight x plus straight y space end cell cell straight x plus 2 straight y space end cell cell straight x space end cell end table close vertical bar Applying space straight R subscript 1 rightwards arrow straight R subscript 1 space plus straight R subscript 2 space plus space straight R subscript 3 space we space get colon open vertical bar table row cell 3 left parenthesis straight x plus straight y right parenthesis end cell cell 3 left parenthesis straight x plus straight y right parenthesis end cell cell 3 left parenthesis straight x plus straight y right parenthesis end cell row cell straight x plus 2 straight y end cell straight x cell straight x plus straight y end cell row cell straight x plus straight y end cell cell straight x plus 2 straight y end cell straight x end table close vertical bar space equals space 3 left parenthesis straight x plus straight y right parenthesis space open vertical bar table row 1 1 1 row cell straight x plus 2 straight y end cell straight x cell straight x plus straight y end cell row cell straight x plus straight y end cell cell straight x plus 2 straight y end cell straight x end table close vertical bar Applying space straight C subscript 1 rightwards arrow straight C subscript 1 minus straight C subscript 2 space and space straight C subscript 2 space rightwards arrow straight C subscript 2 minus straight C subscript 3 space we space get colon 3 left parenthesis straight x plus straight y right parenthesis space open vertical bar table row 0 0 1 row cell 2 straight y end cell cell negative straight y end cell cell straight x plus straight y end cell row cell negative straight y end cell cell 2 straight y end cell straight x end table close vertical bar equals space 3 left parenthesis straight x plus straight y right parenthesis open vertical bar table row cell 2 straight y end cell cell negative straight y end cell row cell negative straight y end cell cell 2 straight y end cell end table close vertical bar space equals space 3 left parenthesis straight x plus straight y right parenthesis left parenthesis 4 straight y squared minus straight y squared right parenthesis space equals space 9 straight y squared space left parenthesis straight x plus straight y right parenthesis space RHS equals space LHS Hence space Proved

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