Evaluate the determinants

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

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Solution

Let space space space increment space equals space open square brackets table row 3 cell space space minus 4 end cell cell space space space space 5 end cell row 1 cell space space space space 1 end cell cell space minus 2 end cell row 2 cell space space space space 3 end cell cell space space space 1 end cell end table close square brackets space space equals 3 space open vertical bar table row 1 cell space space minus 2 end cell row 3 cell space space space space 1 end cell end table close vertical bar space minus left parenthesis negative 4 right parenthesis space space open vertical bar table row 1 cell space space minus 2 end cell row 3 cell space space space space 1 end cell end table close vertical bar space plus 5 space open vertical bar table row 1 cell space space space space space 0 end cell row 2 cell space space space space space 3 end cell end table close vertical bar space space space space space space space space space space space space space equals space 3 left parenthesis 1 plus 6 right parenthesis plus 4 left parenthesis 1 plus 45 right parenthesis plus 5 left parenthesis 3 minus 2 right parenthesis space space space space space space space space space space space space space equals space 3 space left parenthesis 7 right parenthesis plus 4 left parenthesis 5 right parenthesis plus 5 left parenthesis 1 right parenthesis equals 21 plus 20 plus 5 equals 46 space space space space space space space space

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