Find the inverse of $open square brackets table row cell negative 1 end cell cell space space space space 5 end cell row cell negative 3 end cell cell space space space space 2 end cell end table close square brackets$

Solution

Let A = $open square brackets table row cell negative 1 end cell cell space space space 5 end cell row cell negative 3 end cell cell space space space 2 end cell end table close square brackets$
$therefore space space open vertical bar straight A close vertical bar space space equals open square brackets table row cell negative 1 end cell cell space space space space 5 end cell row cell negative 3 end cell cell space space space 2 end cell end table close square brackets space equals space minus 2 plus 15 space equals space 13 space not equal to space 0 therefore space space space space space straight A to the power of negative 1 end exponent space exits. space space space space space space space space space straight A subscript 11 space equals space 2 comma space space space straight A subscript 12 space equals space 3 comma space space space straight A subscript 21 space equals space minus 5 comma space space straight A subscript 22 space equals space minus 1 adj space. straight A space equals space open square brackets table row cell straight A subscript 11 end cell cell straight A subscript 12 end cell row cell straight A subscript 21 end cell cell straight A subscript 22 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell space 2 end cell cell space space space 3 end cell row cell negative 5 end cell cell space space minus 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 2 cell space minus 5 end cell row 3 cell negative 1 end cell end table close square brackets straight A to the power of negative 1 end exponent space equals space fraction numerator adj. space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space 1 over 13 open square brackets table row 2 cell space minus 5 end cell row 3 cell negative 1 end cell end table close square brackets.$

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Determinants

Find the inverse of $open square brackets table row cell negative 1 end cell cell space space space space 5 end cell row cell negative 3 end cell cell space space space space 2 end cell end table close square brackets$

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