If A = $open square brackets table row 2 cell space space 0 end cell cell space space minus 1 end cell row 5 cell space space 1 end cell cell space space space space 0 end cell row 0 cell space space 1 end cell cell space space space space 3 end cell end table close square brackets comma$ prove that A^–1 = A^–2 – 6 A + 111.

Solution

Here $straight A space equals space open square brackets table row 2 cell space space space 0 end cell cell space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space 0 end cell row 0 cell space space space 1 end cell cell space space space space 3 end cell end table close square brackets$
$therefore space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 2 cell space space space 0 end cell cell negative 1 end cell row 5 cell space space 1 end cell cell space space space 0 end cell row 0 cell space space 1 end cell cell space space 3 end cell end table close vertical bar space equals space 2 space open vertical bar table row 1 cell space space space 0 end cell row 1 cell space space 3 end cell end table close vertical bar space minus space 0 space open vertical bar table row 5 cell space space 0 end cell row 0 cell space space 3 end cell end table close vertical bar space plus space left parenthesis negative 1 right parenthesis space open vertical bar table row 5 cell space space 1 end cell row 0 cell space space 1 end cell end table close vertical bar space space space equals space 2 left parenthesis 3 minus 0 right parenthesis minus 0 minus left parenthesis 5 minus 0 right parenthesis space equals space 6 minus 5 space equals space 1 space not equal to space 0 therefore space space space straight A to the power of negative 1 end exponent space exists$
Co-factors of the elements of first row of | A | are
$open vertical bar table row 1 cell space space space 0 end cell row 1 cell space space space 3 end cell end table close vertical bar comma space space space minus open vertical bar table row 5 cell space space space 0 end cell row 0 cell space space space 3 end cell end table close vertical bar comma space space space space space open vertical bar table row 5 cell space space space 1 end cell row 0 cell space space space 1 end cell end table close vertical bar$
i.e. 3, – 15, 5 respectively
Co-factors of the elements of second row of | A | are
$negative open vertical bar table row 0 cell space space space minus 1 end cell row 1 cell space space space space space 3 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space space space minus 1 end cell row 0 cell space space space space space space 3 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space space space 0 end cell row 0 cell space space space space space 1 end cell end table close vertical bar$
i.e. – 1, 6, – 2 respectively
Co-factors of the elements of third row of | A | are
$open vertical bar table row 0 cell space space space minus 1 end cell row 1 cell space space space space space 0 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space minus 1 end cell row 5 cell space space space space space 0 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space space space space 0 end cell row 5 cell space space space space space 1 end cell end table close vertical bar$

i.e. 1, – 5, 2 respectively
$therefore space space space adj. space straight A space equals space open square brackets table row cell space space 3 end cell cell space space space minus 15 end cell cell space space space space 5 end cell row cell negative 1 end cell cell space space space space space 6 end cell cell negative 2 end cell row 1 cell space space space minus 5 end cell cell space space 2 end cell end table close square brackets to the power of apostrophe space equals space space open square brackets table row cell space space 3 end cell cell space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space 6 end cell cell space minus 5 end cell row cell space space 5 end cell cell negative 2 end cell cell space space space space space 2 end cell end table close square brackets 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 over 1 open square brackets table row cell space space space space 3 end cell cell space space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space space 6 end cell cell space space minus 5 end cell row cell space space space space space 5 end cell cell space space space minus 2 end cell cell space space space space space 2 end cell end table close square brackets space equals space open square brackets table row cell space space space space 3 end cell cell space space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space space 6 end cell cell space space minus 5 end cell row cell space space space space space 5 end cell cell space space space minus 2 end cell cell space space space space space 2 end cell end table close square brackets straight A squared space equals space open square brackets table row 2 cell space space space space 0 end cell cell space space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space space 0 end cell row 0 cell space space space space 1 end cell cell space space space space space space 3 end cell end table close square brackets space open square brackets table row 2 cell space space space space 0 end cell cell space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space space 0 end cell row 0 cell space space 1 end cell cell space space space 3 end cell end table close square brackets space space space equals space open square brackets table row cell 4 plus 0 plus 0 end cell cell space space space 0 plus 0 minus 1 end cell cell space space minus 2 plus 0 minus 3 end cell row cell 10 plus 5 plus 0 end cell cell 0 plus 1 plus 0 end cell cell negative 5 plus 0 plus 0 end cell row cell 0 plus 5 plus 0 end cell cell 0 plus 1 plus 3 end cell cell 0 plus 0 plus 9 end cell end table close square brackets space equals space open square brackets table row 4 cell space space minus 1 end cell cell space space space minus 5 end cell row 15 cell space space space space 1 end cell cell space space minus 5 end cell row 5 cell space space space 4 end cell cell space space space 9 end cell end table close square brackets straight R. straight H. straight S space equals space straight A squared minus 6 straight A space plus space 11 space straight I space space space space space space space space space space space space space space space$
$equals space open square brackets table row 4 cell space space minus 1 end cell cell space space space minus 5 end cell row 15 cell space space space space space 1 end cell cell space space minus 5 end cell row 5 cell space space space space 4 end cell cell space space space space space 9 end cell end table close square brackets minus 6 space open square brackets table row 2 cell space space space 0 end cell cell space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space space 0 end cell row 0 cell space space space 1 end cell cell space space space space space 3 end cell end table close square brackets space plus space 11 space open square brackets table row 1 cell space space space 0 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 0 cell space space 0 end cell cell space space 1 end cell end table close square brackets space equals space open square brackets table row 4 cell space space minus 1 end cell cell space space minus 5 end cell row 15 cell space space space space space 1 end cell cell space minus 5 end cell row 5 cell space space space 4 end cell cell space space space space 9 end cell end table close square brackets space plus space open square brackets table row cell negative 12 end cell cell space space space space 0 end cell cell space space space 6 end cell row cell negative 30 end cell cell space minus 6 end cell cell space space space 0 end cell row 0 cell space space minus 6 end cell cell space space space minus 18 end cell end table close square brackets space plus space space open square brackets table row 11 cell space 0 end cell cell space space 0 end cell row 0 cell space space 11 end cell cell space space 0 end cell row 0 0 cell space space 11 end cell end table close square brackets equals space open square brackets table row cell 4 minus 12 plus 11 end cell cell space space space minus 1 plus 0 plus 0 end cell cell space space space space space minus 5 plus 6 plus 0 end cell row cell 15 minus 30 plus 0 end cell cell space space 1 minus 6 plus 11 end cell cell space space minus 5 plus 0 plus 0 end cell row cell 5 plus 0 plus 0 end cell cell space 4 minus 6 plus 0 end cell cell space space space space space 9 minus 18 plus 11 end cell end table close square brackets space equals space open square brackets table row cell space space 3 end cell cell space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space 6 end cell cell space minus 5 end cell row cell space space space 5 end cell cell space minus 2 end cell cell space space space space 2 end cell end table close square brackets equals space straight A squared space equals space straight R. straight H. straight S.$

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