Solve the equation
$open vertical bar table row cell straight x plus straight a end cell cell space straight x end cell cell space straight x end cell row straight x cell space straight x plus straight a end cell cell space straight x end cell row straight x straight x cell space straight x plus straight a end cell end table close vertical bar space equals space 0 comma space space space straight a not equal to space 0.$

Solution

The given equation is
$open vertical bar table row cell straight x plus straight a end cell cell space straight x end cell cell space straight x end cell row straight x cell space straight x plus straight a end cell cell space straight x end cell row straight x straight x cell space straight x plus straight a end cell end table close vertical bar space equals space 0$
$rightwards double arrow space open vertical bar table row cell 3 straight x plus straight a end cell cell space 3 straight x plus straight a end cell cell 3 straight x plus straight a end cell row straight x cell straight x plus straight a end cell straight x row straight x straight x cell straight x plus straight a end cell end table close vertical bar space equals space 0 comma space space by space straight R subscript 1 space rightwards arrow space straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3 rightwards double arrow space left parenthesis 3 straight x plus straight a right parenthesis thin space open vertical bar table row 1 cell space space space 1 end cell cell space 1 end cell row straight x cell space space space space straight x plus straight a end cell cell space straight x end cell row straight x straight x cell space straight x plus straight a end cell end table close vertical bar space equals space 0 rightwards double arrow space left parenthesis 3 straight x plus straight a right parenthesis thin space open vertical bar table row 1 cell space space space 0 end cell cell space space space 0 end cell row straight x cell space space straight a end cell cell space space 0 end cell row straight x cell space space 0 end cell cell space space straight a end cell end table close vertical bar space equals space 0 comma space space by space straight C subscript 2 space rightwards arrow space straight C subscript 2 space minus space straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 rightwards double arrow space space space left parenthesis 3 straight x plus straight a right parenthesis thin space left square bracket left parenthesis 1 right parenthesis space left parenthesis straight a space left parenthesis straight a right parenthesis right square bracket space equals 0 space space space space space space space space space space space space space space space space left square bracket Product space of space diagonal space elements right square bracket rightwards double arrow space space space space space straight a squared left parenthesis 3 straight x plus straight a right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space 3 straight x plus straight a space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space straight a not equal to space 0 close square brackets rightwards double arrow space space space space space space space straight x equals negative straight a over 3$

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