If $straight A space equals space open square brackets table row 0 cell space space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets comma$ show that A^–1 = A.

Solution

straight A space equals space open square brackets table row 0 cell space space space 0 end cell cell space space space 1 end cell row 0 cell space space space 1 end cell cell space space space space 0 end cell row 0 cell space space space space 0 end cell cell space space 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 0 cell space space space 0 end cell cell space space space space 1 end cell row 0 cell space space 1 end cell cell space space space 0 end cell row 1 cell space space space 0 end cell cell space space space 0 end cell end table close vertical bar space equals 1 open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space space 0 end cell end table close vertical bar space equals space 1 left parenthesis 0 minus 1 right parenthesis space equals space minus 1 not equal to space space 0 space space space space rightwards double arrow space space straight A to the power of negative 1 end exponent space exists. space

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

open vertical bar table row 1 cell space space 0 end cell row 0 cell space 0 end cell end table close vertical bar comma space space minus open vertical bar table row 0 cell space space space space 0 end cell row 1 cell space space space space 0 end cell end table close vertical bar comma space space open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space 0 end cell end table close vertical bar

i.e. 0, 0, – 1 respectively
Co-factors of the elements of the second row of | A | are

negative open vertical bar table row 0 cell space space 1 end cell row 0 cell space space 0 end cell end table close vertical bar comma space space open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space space 0 end cell end table close vertical bar comma space space minus open vertical bar table row 0 cell space space 0 end cell row 1 cell space space 0 end cell end table close vertical bar

i.e. 0,–1, 0, respectively.
Co-factors of the elements of the third row of | A | are

open vertical bar table row 0 cell space space 1 end cell row 1 cell space 0 end cell end table close vertical bar comma space space minus open vertical bar table row 0 cell space space space space 1 end cell row 0 cell space space space space 0 end cell end table close vertical bar comma space space open vertical bar table row 0 cell space space space 0 end cell row 0 cell space space 1 end cell end table close vertical bar

i.e., – 1, 0, 0 respectively.

therefore space space space adj. space straight A space equals space open square brackets table row 0 cell space space 0 end cell cell space space space minus 1 end cell row 0 cell space minus 1 end cell cell space space space space space space 0 end cell row cell negative 1 end cell cell space space 0 end cell cell space space space space space space 0 end cell end table close square brackets to the power of apostrophe space equals space space open square brackets table row cell space 0 end cell cell space space space space 0 end cell cell space space space minus 1 end cell row cell space space 0 end cell cell space space minus 1 end cell cell space space space space space 0 end cell row cell negative 1 end cell cell space space space space 0 end cell cell space space space space 0 end cell end table close square brackets Now 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 fraction numerator 1 over denominator negative 1 end fraction open square brackets table row 0 cell space space space space 0 end cell cell space space space minus 1 end cell row 0 cell space space minus 1 end cell cell space space space space space 0 end cell row cell negative 1 end cell cell space space space space 0 end cell cell space space space space 0 end cell end table close square brackets space equals space open square brackets table row 0 cell space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets space equals space straight A therefore space space space straight A to the power of negative 1 end exponent space equals space straight A.

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