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Determinants

Question
CBSEENMA12034659
Wired Faculty App

If straight A space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 2 cell space minus 1 end cell cell space space 0 end cell row 1 cell space space space space 0 end cell cell space space 0 end cell end table close square brackets comma find A–1 and show that A–1 = A2.

Solution
straight A space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 2 cell space space minus 1 end cell cell space space 0 end cell row 1 cell space space space space space 0 end cell cell space space 0 end cell end table close square brackets open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space minus 1 end cell cell space space 1 end cell row 2 cell space minus 1 end cell cell space space 0 end cell row 1 cell space space space 0 end cell cell space space 0 end cell end table close vertical bar space equals space 1 open vertical bar table row 2 cell space space minus 1 end cell row 1 cell space space space space 0 end cell end table close vertical bar space equals space 0 plus 1 space equals space 1 space not equal to 0 space space space space rightwards double arrow 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 cell negative 1 end cell cell space space space 0 end cell row 0 cell 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 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 2 cell space space minus 1 end cell row 1 cell space space space space space 0 end cell end table close vertical bar
i.e.   0, – 1, – 1 respectively.
Co-factors of the elements of third row of | A | are
open vertical bar table row cell negative 1 end cell cell space space 1 end cell row cell negative 1 end cell cell space space 0 end cell end table close vertical bar comma space space space space minus open vertical bar table row 1 cell space space space space space 1 end cell row 2 cell space space space space space 0 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space space space minus 1 end cell row 2 cell space space space space minus 1 end cell end table close vertical bar
i.e.. 1, 2, 1 respectively
therefore space space space adj. space straight A space equals space open square brackets table row 0 cell space space space space space 0 end cell cell space space space space space space 1 space end cell row 0 cell space space minus 1 end cell cell space space minus 1 end cell row 1 cell space space space space space 2 end cell cell space space space space space 1 end cell end table close square brackets 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 negative 1 end cell cell space space space 2 end cell row 1 cell negative 1 end cell cell space space 1 end cell end table close square brackets 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 open square brackets table row 0 cell space space space space space 0 end cell cell space space 1 space end cell row 0 cell space minus 1 end cell cell space space 2 end cell row 1 cell space minus 1 end cell cell space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space space space space space straight A squared space equals space open square brackets table row 1 cell space space space minus 1 end cell cell space space 1 end cell row 2 cell space space minus 1 end cell cell space space 0 end cell row 1 cell space space space space 0 end cell cell space space 0 end cell end table close square brackets space space space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 2 cell space space minus 1 end cell cell space space 0 end cell row 1 cell space space space 0 end cell cell space space 0 end cell end table close square brackets space equals space open square brackets table row cell 1 minus 2 plus 1 end cell cell space space space space space minus 1 plus 1 plus 0 end cell cell space space space space 1 plus 0 plus 0 end cell row cell 2 minus 2 plus 0 end cell cell space space space minus 2 plus 1 plus 0 end cell cell space space space space 2 plus 0 plus 0 end cell row cell 1 plus 0 plus 0 end cell cell space space space minus 1 plus 0 plus 0 end cell cell space space 1 plus 0 plus 0 end cell end table close square brackets therefore space space space space space space space space space straight A squared space equals space open square brackets table row 0 cell space space space space 0 end cell cell space space space space 1 end cell row 0 cell space space minus 1 end cell cell space space space space 2 end cell row 1 cell space minus 1 end cell cell space space space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space space space space space space space space space space space space

From (1) and (2),
A–1 = A2

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