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Determinants

Question

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Find the inverse of $open square brackets table row 2 cell space space space minus 2 end cell row 4 cell space space space space space space 3 end cell end table close square brackets$

Solution

straight A space equals space open square brackets table row 2 cell negative 2 end cell row 4 cell space space space 3 end cell end table close square brackets

therefore space space space space space open vertical bar straight A close vertical bar space equals open square brackets table row 2 cell space space space minus 2 end cell row 4 cell space space space space space space 3 end cell end table close square brackets space equals space 6 plus 8 space equals space 14 space not equal to 0 therefore space space space space space straight A to the power of negative 1 end exponent space exists. space space space space space space space space space space straight A subscript 11 space equals space 3 comma space space space straight A subscript 12 space equals space minus 4 comma space space straight A subscript 21 space equals space 2 comma space space straight A subscript 22 space equals space 2 space space space space space space space space space adj. space straight A space equals space open square brackets table row cell straight A subscript 11 space end subscript end cell cell space straight A subscript 12 end cell row cell straight A subscript 13 end cell cell space straight A subscript 14 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 3 cell space space space space minus 4 end cell row 2 cell space space space space space space space 2 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell space space 3 end cell cell space space 2 end cell row cell negative 4 end cell cell space space 2 end cell end table close square brackets space space space space space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator adj. space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space 1 over 14 open square brackets table row cell space space space 3 end cell cell space space 2 end cell row cell negative 4 end cell cell space space 2 end cell end table close square brackets

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