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Determinants

Question
CBSEENMA12034662
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If straight A space equals space open square brackets table row 3 cell space space minus 3 end cell cell space space space 4 end cell row 2 cell space space minus 3 end cell cell space space space 4 end cell row 0 cell space space minus 1 end cell cell space space space 1 end cell end table close square brackets comma  show that A4 = I. Hence find A–1

Solution
straight A space equals space open square brackets table row 3 cell space space space minus 3 end cell cell space space 4 end cell row 2 cell space space minus 3 end cell cell space space 4 end cell row 0 cell space minus 1 end cell cell space space 1 end cell end table close square brackets
Now straight A squared space equals space open square brackets table row 3 cell space space minus 3 end cell cell space space 4 end cell row 2 cell space space minus 3 end cell cell space space 4 end cell row 0 cell negative 1 end cell cell space space 1 end cell end table close square brackets space open square brackets table row 3 cell space space space minus 3 end cell cell space space space 4 end cell row 2 cell space space space minus 3 end cell cell space space space 4 end cell row 0 cell space minus 1 end cell cell space space space 1 end cell end table close square brackets space space space space space space space
          equals space open square brackets table row cell 9 minus 6 plus 0 end cell cell space space minus 9 plus 9 minus 4 end cell cell space space 12 minus 12 plus 4 end cell row cell 6 minus 6 plus 0 end cell cell space minus 6 plus 9 minus 4 end cell cell 8 minus 12 plus 4 end cell row cell 0 minus 2 plus 0 end cell cell 0 plus 3 minus 1 end cell cell 0 minus 4 plus 1 end cell end table close square brackets space equals space open square brackets table row 3 cell space space minus 4 end cell cell space space space space 4 end cell row 0 cell space minus 1 end cell cell space space space 0 end cell row cell negative 2 end cell cell space 2 end cell cell space minus 3 end cell end table close square brackets straight A cubed space equals space open square brackets table row 3 cell space space minus 4 end cell cell space space space space space space 4 end cell row 0 cell space space minus 1 end cell cell space space space space space 0 end cell row cell negative 2 end cell cell space space 2 end cell cell space space minus 3 end cell end table close square brackets space space open square brackets table row 3 cell space minus 3 end cell cell space space 4 end cell row 2 cell negative 3 end cell cell space space 4 end cell row 0 cell negative 1 end cell cell space space 1 end cell end table close square brackets space space space space space space space equals space open square brackets table row cell 9 minus 8 plus 0 end cell cell space space space minus 9 plus 12 minus 4 end cell cell space space 12 minus 6 plus 4 end cell row cell 0 minus 2 plus 0 end cell cell 0 plus 3 plus 0 end cell cell 0 minus 4 plus 0 end cell row cell negative 6 plus 4 plus 0 end cell cell 6 minus 6 plus 3 end cell cell negative 8 plus 8 minus 3 end cell end table close square brackets space equals space open square brackets table row cell space space 1 end cell cell space space minus 1 end cell cell space space space space 0 end cell row cell negative 2 end cell cell space space space space space 3 end cell cell space minus 4 end cell row cell negative 2 end cell cell space space space space 3 end cell cell space minus 3 end cell end table close square brackets space space space space space space space space space space... left parenthesis 2 right parenthesis straight A to the power of 4 space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space space space 0 end cell row cell negative 2 end cell cell space space space 3 end cell cell negative 4 end cell row cell negative 2 end cell cell space space 3 end cell cell negative 3 end cell end table close square brackets space space space open square brackets table row 3 cell space space minus 3 end cell cell space space space 4 end cell row 2 cell space space minus 3 end cell cell space space 4 end cell row 0 cell space space minus 1 end cell cell space space 1 end cell end table close square brackets space space space space space space equals space open square brackets table row cell 3 minus 2 plus 0 end cell cell space space space minus 3 plus 3 plus 0 end cell cell space space space space 4 plus 4 plus 0 end cell row cell negative 6 plus 6 plus 0 end cell cell 6 minus 9 plus 4 end cell cell negative 8 plus 12 minus 4 end cell row cell negative 6 plus 6 plus 0 end cell cell 6 minus 9 plus 3 end cell cell negative 8 plus 12 minus 3 end cell end table close square brackets space equals space open square brackets table row 1 cell space space 0 end cell cell space space 0 end cell row 0 cell 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 therefore space space space space space straight A to the power of 4 space equals space space straight I space space space space rightwards double arrow space space space space straight A to the power of 4 straight A to the power of negative 1 end exponent space space equals IA to the power of negative 1 end exponent space space space space rightwards double arrow space space space space space straight A cubed space equals space straight A to the power of negative 1 end exponent space rightwards double arrow space space space space space space straight A to the power of negative 1 end exponent space space equals space open square brackets table row 1 cell space space space minus 1 end cell cell space space space space space space 0 end cell row cell negative 2 end cell cell space space space space space 3 end cell cell space space minus 4 end cell row cell negative 2 end cell cell space space space space 3 end cell cell space space minus 3 end cell end table close square brackets space space
              

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