Find the value of the determinant

$open vertical bar table row 1 cell space minus 3 end cell cell space space space 2 end cell row 4 cell space minus 1 end cell cell space space space 2 end cell row 3 cell space space space space space 5 end cell cell space space space 2 end cell end table close vertical bar$

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Solution

Let space space increment space space equals space space open vertical bar table row 1 cell space minus 3 end cell cell space space space 2 end cell row 4 cell space minus 1 end cell cell space space space 2 end cell row 3 cell space space space space space 5 end cell cell space space space 2 end cell end table close vertical bar space space space space space space space space space space space space space space equals space space 1 space open square brackets table row cell negative 1 end cell cell space space 2 end cell row 5 cell space space 2 end cell end table close square brackets space minus left parenthesis negative 3 right parenthesis space open vertical bar table row 4 cell space space 2 end cell row 3 cell space space 2 end cell end table close vertical bar plus 2 space open vertical bar table row 4 cell space minus 1 end cell row 3 cell space space space space 5 end cell end table close vertical bar space space space space space space space space space space space space space space equals space space 1 space space left parenthesis negative 2 minus 10 right parenthesis plus 3 left parenthesis 8 minus 6 right parenthesis plus 2 left parenthesis 20 plus 3 right parenthesis space space space space space space space space space space space space space space equals space minus 12 plus 6 plus 46 equals 40 space space space space

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