Find the inverse of the matrix:
$open square brackets table row 1 cell space space 2 end cell cell space space 2 end cell row 2 cell space 1 end cell cell space 2 end cell row 2 cell space 2 end cell cell space 1 end cell end table close square brackets$

Solution

straight A space equals space open square brackets table row 1 cell space space 2 end cell cell space space 2 end cell row 2 cell space space 1 end cell cell space space 2 end cell row 2 cell space space 2 end cell cell space space 1 end cell end table close square brackets therefore space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space 2 end cell cell space space space 2 end cell row 2 cell space space 1 end cell cell space space space 2 space end cell row 2 cell space space 2 end cell cell space space 1 end cell end table close vertical bar space equals space 1 open vertical bar table row 1 cell space space 2 end cell row 2 cell space space space 1 end cell end table close vertical bar minus 2 open vertical bar table row 2 cell space space 2 end cell row 2 cell space space 1 end cell end table close vertical bar plus 2 space open vertical bar table row 2 cell space space 1 end cell row 2 cell 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 1 left parenthesis 1 minus 4 right parenthesis minus 2 left parenthesis 2 minus 4 right parenthesis plus 2 left parenthesis 4 minus 2 right parenthesis space equals negative 3 plus 4 plus 4 space equals space 5 therefore space space space straight A to the power of negative 1 end exponent space exists

Co-factors of the elements of the first row of | A | are

open vertical bar table row 1 cell space space 2 end cell row 2 cell space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space 2 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space space open vertical bar table row 2 cell space space space space 1 end cell row 2 cell space space space space 2 end cell end table close vertical bar

i.e. – 3, 2, 2 respectively
Co-factors of the elements of the second row of | A | are

negative open vertical bar table row 2 cell space space space 2 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space open vertical bar table row 1 cell space space space 2 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 1 cell space space 2 end cell row 2 cell space space 2 end cell end table close vertical bar

i.e. 2, – 3, 2 respectively.
Co-factors of the elements of the third row of | A | are

open vertical bar table row 2 cell space space space 2 end cell row 1 cell space space space 2 end cell end table close vertical bar comma space space space minus open vertical bar table row 1 cell space space space 2 end cell row 2 cell space space 2 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space 2 end cell row 2 cell space space 1 end cell end table close vertical bar

i.e., 2, 2, – 3 respectively

therefore space space space adj. space straight A space equals space open square brackets table row cell negative 3 end cell cell space space 2 end cell cell space space space space 2 end cell row 2 cell space minus 3 end cell cell space space space space 2 end cell row 2 2 cell space minus 3 end cell end table close square brackets to the power of apostrophe space equals open square brackets table row cell negative 3 end cell cell space space space space space 2 end cell cell space space space space space 2 end cell row cell space space 2 end cell cell space minus 3 end cell cell space space space space space 2 end cell row cell space 2 end cell cell space space space 2 end cell cell space minus 3 end cell end table close square brackets Now comma 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 fifth open square brackets table row cell negative 3 end cell cell space space space space 2 end cell cell space space space space space 2 end cell row cell space 2 end cell cell space minus 3 end cell cell space space space space space space 2 end cell row 2 cell space space space 2 end cell cell space space minus 3 end cell end table close square brackets

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