Question

Find $straight A to the power of negative 1 end exponent$ if $straight A space equals space open square brackets table row 0 cell space space space space 1 end cell cell space space space space 1 end cell row 1 cell space space space space 0 end cell cell space space space 1 end cell row 1 cell space space space space 1 end cell cell space space space 0 end cell end table close square brackets$ Also show that $straight A to the power of negative 1 end exponent space space equals fraction numerator straight A squared minus 3 straight I over denominator 2 end fraction.$

Solution

straight A space equals space open square brackets table row 0 cell space space 1 end cell cell space 1 space end cell row 1 cell space space 0 end cell 1 row 1 cell space 1 end cell 0 end table close square brackets

therefore space space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 0 cell space space space space 1 end cell cell space space space 1 end cell row 1 cell space space space 0 end cell cell space space space space 1 end cell row 1 cell space space space 1 end cell cell space space space 0 end cell end table close vertical bar space equals space 0 space open vertical bar table row 0 cell space space 1 end cell row 1 cell space space 0 end cell end table close vertical bar space minus space 1 space open vertical bar table row 1 cell space space 1 end cell row 1 cell space space 0 end cell end table close vertical bar space plus space 1 space open vertical bar table row 1 cell space space 0 end cell row 1 cell space space 1 end cell end table close vertical bar space space space space space space space space space space space space space space space equals space 0 left parenthesis 0 minus 1 right parenthesis space minus 1 space left parenthesis 0 minus 1 right parenthesis space plus space 1 left parenthesis 1 minus 0 right parenthesis space equals space 0 plus 1 plus 1 space equals space 2 space not equal to 0 therefore space space space straight A to the power of negative 1 end exponent space exists.

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

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

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

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

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

open vertical bar table row 1 cell space space 1 end cell row 0 cell space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space 1 end cell end table close vertical bar comma space space open vertical bar table row 0 cell space space 1 end cell row 1 cell space space 0 end cell end table close vertical bar space space space straight i. straight e. space space space space space 1 comma space 1 comma space minus 1 space respectively.

therefore space space space space adj space straight A space equals space open square brackets table row cell negative 1 end cell 1 cell space space 1 end cell row cell space space 1 end cell cell negative 1 end cell cell space space 1 end cell row cell space space 1 end cell 1 cell negative 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 1 end cell cell space space space space 1 end cell cell space space space space space 1 end cell row 1 cell negative 1 end cell cell space space space space space space 1 end cell row 1 cell space space 1 end cell cell space space minus 1 end cell end table close square brackets

space space space space space 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 half open square brackets table row cell negative 1 end cell cell space space space 1 end cell cell space space space 1 end cell row 1 cell negative 1 end cell cell space space space space 1 space end cell row 1 cell space 1 end cell cell negative 1 end cell end table close square brackets space space space space space space space space space space space space straight A squared space equals space open square brackets table row 0 cell space space 1 end cell cell space space space 1 end cell row 1 cell space space 0 end cell cell space space space 1 end cell row 1 cell space space 1 end cell cell space space space 0 end cell end table close square brackets space space open square brackets table row 0 cell space space 1 end cell cell space space 1 end cell row 1 cell space space 0 end cell cell space space 1 end cell row 1 cell space 1 end cell cell space space 0 end cell end table close square brackets space equals space open square brackets table row 2 cell space space 1 end cell cell space space space 1 end cell row 1 cell space space 2 end cell cell space space 1 end cell row 1 cell space space 1 end cell cell space space 2 end cell end table close square brackets therefore space space space straight A squared minus 3 straight I space equals space open square brackets table row 2 cell space space space 1 end cell cell space space 1 end cell row 1 cell space space 2 end cell cell space space 1 end cell row 1 cell space space 1 end cell cell space space 2 end cell end table close square brackets space minus space space 3 open square brackets table row 1 cell space 0 end cell cell space 0 end cell row 0 cell space 1 space end cell cell space 0 end cell row 0 cell space 0 end cell cell space 1 end cell end table close square brackets space equals space open square brackets table row cell negative 1 end cell cell space 1 end cell cell space space space space 1 end cell row 1 cell negative 1 end cell cell space space space space 1 end cell row 1 cell space space 1 end cell cell space space minus 1 end cell end table close square brackets therefore space space space space fraction numerator straight A squared minus 3 straight I over denominator 2 end fraction space equals space 1 half open square brackets table row cell negative 1 end cell cell space space space 1 end cell cell space space space space 1 end cell row cell space space 1 end cell cell space minus 1 end cell cell space space space space 1 end cell row cell space space 1 end cell cell space space 1 end cell cell space space space space 1 end cell end table close square brackets therefore space space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator straight A squared space minus space 3 straight I over denominator 2 end fraction

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