Write the element $straight a subscript 12$ of the matrix $straight A space equals space open square brackets straight a subscript ij close square brackets subscript 2 cross times 2 end subscript comma$ whose elements $straight a subscript ij$ are given by $aij space equals space straight e to the power of 2 ix end exponent space sin space jx.$

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straight A to the power of negative 1 end exponent space equals space open square brackets table row cell fraction numerator negative 3 over denominator 5 end fraction end cell cell space 2 over 5 end cell cell space 2 over 5 end cell row cell 2 over 5 end cell cell fraction numerator negative 3 over denominator 5 end fraction end cell cell 2 over 5 end cell row cell 2 over 5 end cell cell 2 over 5 end cell cell fraction numerator negative 3 over denominator 5 end fraction end cell end table close square brackets

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$If space straight A space equals space open vertical bar table row 2 cell space space 0 end cell cell negative 1 end cell row 5 cell space 1 end cell cell space 0 end cell row 0 cell space 1 end cell cell space 3 end cell end table close vertical bar comma space then space find thin space straight A to the power of negative 1 end exponent space using space elementary space row space operations.$ $WiredFaculty$

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Using the properties of determinants, solve the following for x:

$open vertical bar table row cell straight x plus 2 end cell cell space space straight x plus 6 end cell cell space straight x minus 1 end cell row cell straight x plus 6 end cell cell straight x minus 1 end cell cell straight x plus 2 end cell row cell straight x minus 1 end cell cell straight x plus 2 end cell cell straight x plus 6 end cell end table close vertical bar space equals space 0$

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$Let space increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space straight x plus 6 end cell cell space straight x minus 1 end cell row cell straight x plus 6 end cell cell straight x minus 1 end cell cell straight x plus 2 end cell row cell straight x minus 1 end cell cell straight x plus 2 end cell cell straight x plus 6 end cell end table close vertical bar space equals space 0$
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