Question

Solve:
$open vertical bar table row cell straight x plus 1 end cell 2 3 row 3 cell straight x plus 2 end cell 1 row 1 2 cell straight x plus 3 end cell end table close vertical bar space equals 0$

Solution

The given equation is
$open vertical bar table row cell straight x plus 1 end cell 2 cell space space space 3 end cell row 3 cell straight x plus 2 end cell cell space space space space 1 end cell row 1 2 cell straight x plus 3 end cell end table close vertical bar space equals space 0$
or $open vertical bar table row cell straight x plus 6 end cell cell space 2 end cell 3 row cell straight x plus 6 end cell cell space straight x plus 2 end cell 1 row cell straight x plus 6 end cell 2 cell straight x plus 3 end cell end table close vertical bar space equals space 0 comma space space by space straight C subscript 1 space rightwards arrow space straight C subscript 1 space plus space straight C subscript 2 plus straight C subscript 3$
or $left parenthesis straight x plus 6 right parenthesis space equals space open vertical bar table row 1 cell space space space space 2 end cell cell space space 3 end cell row 1 cell space space space space straight x plus 2 end cell cell space space 1 end cell row 1 2 cell space space straight x plus 3 end cell end table close vertical bar space equals space 0$
$or space space left parenthesis straight x plus 6 right parenthesis space open vertical bar table row 1 cell space space space 2 end cell cell space space 3 end cell row 0 cell space space space straight x end cell cell space space minus 2 end cell row 0 cell space space 0 end cell cell space space straight x end cell end table close vertical bar space equals space 0 comma space space by space straight R subscript 1 space rightwards arrow space straight R subscript 2 space minus space straight R subscript 1 comma space space space straight R subscript 3 space rightwards arrow space straight R subscript 3 space minus straight R subscript 1 or space space left parenthesis straight x plus 6 right parenthesis thin space left square bracket left parenthesis 1 right parenthesis thin space left parenthesis straight x right parenthesis thin space left parenthesis straight x right parenthesis right square bracket space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets Product space of space diagonal space elements close square brackets or space space space straight x squared left parenthesis straight x plus 6 right parenthesis space equals space 0 therefore space space space space straight x space equals space 0 comma space space 0 comma space space minus 6 therefore space space space space roots space of space given space equation space are space 0 comma space minus 6.$

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Determinants

Solve:
$open vertical bar table row cell straight x plus 1 end cell 2 3 row 3 cell straight x plus 2 end cell 1 row 1 2 cell straight x plus 3 end cell end table close vertical bar space equals 0$

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