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Determinants

Question
CBSEENMA12034674
Wired Faculty App

If straight A space equals space open square brackets table row cell space space 2 end cell cell space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 1 end cell cell space space space space space 2 end cell cell space minus 1 end cell row cell space space 1 end cell cell space space minus 1 end cell cell space space space space 2 end cell end table close square brackets comma
verify that A1 – 6A2 + 9A – 4 I = O and hence find A–1.

Solution
straight A space equals space open square brackets table row 2 cell space space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 1 end cell cell space space space space space 2 end cell cell space space minus 1 end cell row 1 cell space space minus 1 end cell cell space space space 2 end cell end table close square brackets comma
straight A squared space equals space open square brackets table row cell space 2 end cell cell space space space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 1 end cell cell space space space space space space space 2 end cell cell space minus 1 end cell row cell space space 1 end cell cell space space space minus 1 end cell cell space space space space 2 end cell end table close square brackets space open square brackets table row cell space 2 end cell cell space space space minus 1 end cell cell space space space space space space 1 end cell row cell negative 1 end cell cell space space space space space space space 2 end cell cell space space minus 1 end cell row cell space 1 end cell cell space space space minus 1 end cell cell space space space space 2 end cell end table close square brackets space space space space space space space equals open square brackets table row cell 4 plus 1 plus 1 end cell cell space minus 2 minus 2 minus 1 end cell cell space space space 2 plus 1 plus 2 end cell row cell negative 2 minus 2 minus 1 end cell cell space space 1 plus 4 plus 1 end cell cell negative 1 minus 2 minus 2 end cell row cell 2 plus 1 plus 2 end cell cell negative 1 minus 2 minus 2 end cell cell 1 plus 1 plus 4 end cell end table close square brackets space equals space open square brackets table row 6 cell space minus 5 end cell cell space space 5 end cell row cell negative 5 end cell cell space space 6 end cell cell negative 5 end cell row 5 cell negative 5 end cell cell space space 6 end cell end table close square brackets space space space straight A cubed space equals space open square brackets table row cell space space 6 end cell cell space space minus 5 end cell cell space space space space space space space 5 end cell row cell negative 5 end cell cell space space space space space 6 end cell cell space space space minus 5 end cell row cell space 5 end cell cell space space minus 5 end cell cell space space space space space space 6 end cell end table close square brackets space space open square brackets table row 2 cell space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 1 end cell cell space space space space space space 2 end cell cell space space space minus 1 end cell row 1 cell space space space minus 1 end cell cell space space space space space 2 end cell end table close square brackets space space space space equals open square brackets table row 6 cell space space space minus 5 end cell cell space space space space space 5 end cell row cell negative 5 end cell cell space space space space space 6 end cell cell space minus 5 end cell row 5 cell space minus 5 end cell cell space space space space 6 end cell end table close square brackets space open square brackets table row cell space 2 end cell cell space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 1 end cell cell space space space space space 2 end cell cell space minus 1 end cell row 1 cell space space minus 1 end cell cell space space space 2 end cell end table close square brackets space space space space equals space open square brackets table row cell 12 plus 5 plus 5 end cell cell space space space minus 6 minus 10 minus 5 end cell cell space space space 6 plus 5 plus 10 end cell row cell negative 10 minus 6 minus 5 end cell cell space space 5 plus 12 plus 5 end cell cell negative 5 minus 6 minus 10 end cell row cell 10 plus 5 plus 6 end cell cell negative 5 minus 10 minus 6 end cell cell 5 plus 5 plus 12 end cell end table close square brackets space equals space open square brackets table row 22 cell space space minus 21 end cell cell space space 21 end cell row cell negative 21 end cell cell space space space 22 end cell cell negative 21 end cell row 21 cell negative 21 end cell cell space space 22 end cell end table close square brackets
Consider A3 – 6A2 + 9A – 4 I
equals space open square brackets table row cell space space 22 end cell cell space space space space minus 21 end cell cell space space space 21 end cell row cell negative 21 end cell cell space space space space 22 end cell cell negative 21 end cell row cell space space 21 end cell cell negative 21 end cell cell space space 22 end cell end table close square brackets minus 6 space open square brackets table row cell space space 6 end cell cell space space space minus 5 end cell cell space space 5 end cell row cell negative 5 end cell cell space space space space space space space 6 end cell cell negative 5 end cell row cell space space 5 end cell cell space space space minus 5 end cell cell space space 6 end cell end table close square brackets plus 9 space open square brackets table row 2 cell space space minus 1 end cell 1 row cell negative 1 end cell cell space space space space space 2 end cell cell negative 1 end cell row 1 cell negative 1 end cell 2 end table close square brackets minus 4 open square brackets table row 1 cell space space 0 end cell cell space 0 end cell row 0 cell space 1 end cell cell space 0 end cell row 0 cell space 0 end cell 1 end table close square brackets equals space open square brackets table row 22 cell space space space minus 21 end cell cell space space 21 end cell row cell negative 21 end cell cell space space space 22 end cell cell space minus 21 end cell row 21 cell space space minus 21 end cell cell space space space space 22 end cell end table close square brackets space plus space open square brackets table row cell negative 36 end cell cell space 30 end cell cell space minus 30 end cell row 30 cell negative 36 end cell cell space space space 30 end cell row 30 30 cell space space minus 36 end cell end table close square brackets plus space open square brackets table row 18 cell space minus 9 end cell cell space space space space 9 end cell row cell negative 9 end cell cell space space space 18 end cell cell negative 9 end cell row 9 cell negative 9 end cell 18 end table close square brackets plus open square brackets table row cell negative 4 end cell cell space space 0 end cell cell space space space space 0 end cell row 0 cell space minus 4 end cell cell space space space space 2 end cell row 0 cell space 0 end cell cell space minus 4 end cell end table close square brackets equals space open square brackets table row cell 22 minus 36 plus 18 minus 4 end cell cell space space space space space minus 21 plus 30 minus 9 plus 0 end cell cell space space space space space space 21 minus 30 plus 9 plus 0 end cell row cell negative 21 plus 30 minus 9 plus 0 end cell cell space space space space 22 minus 36 plus 18 minus 4 end cell cell space space space minus 21 plus 30 minus 9 plus 0 end cell row cell 21 minus 30 plus 9 plus 0 end cell cell space space minus 21 plus 30 minus 9 plus 0 end cell cell space space space space 22 minus 36 plus 18 minus 4 end cell end table close square brackets equals space space open square brackets table row 0 cell space 0 end cell cell space space 0 end cell row 0 cell space 0 end cell cell space 0 end cell row 0 cell space 0 end cell cell space 0 end cell end table close square brackets space equals space straight O space space space space space space space space space therefore space space space space space space straight A cubed minus 6 straight A squared plus 9 straight A minus 4 straight I space space equals space straight O space space space space space space space space space space rightwards double arrow space space space space 4 space straight I space equals space straight A cubed space minus space 6 straight A squared space plus space 9 straight A space space space space space space space space space rightwards double arrow space space space 4 space straight A to the power of negative 1 end exponent space equals space straight A squared space minus space 6 straight A plus space 9 space straight I space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space 4 straight A to the power of negative 1 end exponent space equals space open square brackets table row cell space space space space 6 end cell cell space space space minus 5 end cell cell space space space space 5 end cell row cell negative 5 end cell cell space space space space space space 6 end cell cell negative 5 end cell row cell space space 5 end cell cell negative 5 end cell cell space space space space 6 end cell end table close square brackets minus 6 space open square brackets table row 2 cell space space minus 1 end cell cell space space 1 end cell row cell negative 1 end cell cell space space space space space 2 end cell cell negative 1 end cell row 1 cell negative 1 end cell 2 end table close square brackets space plus space 9 space open square brackets table row 1 cell space 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 0 end cell cell space 1 end cell end table close square brackets space
           rightwards double arrow space space space 4 straight A to the power of negative 1 end exponent space equals space open square brackets table row cell space space space 6 end cell cell space space minus 5 end cell cell space space space 5 end cell row cell negative 5 end cell cell space space space space space 6 end cell cell negative 5 end cell row cell space space space 5 end cell cell space space minus 5 end cell cell space space space 6 end cell end table close square brackets plus space open square brackets table row cell negative 12 end cell cell space 6 end cell cell space minus 6 end cell row cell space 6 end cell cell negative 12 end cell cell space space space space 6 end cell row cell negative 6 end cell 6 cell negative 12 end cell end table close square brackets plus space open square brackets table row 9 cell space 0 end cell cell space space 0 end cell row 0 cell space 9 end cell cell space 0 end cell row 0 cell space 0 end cell cell space 9 end cell end table close square brackets rightwards double arrow space space space space 4 straight A to the power of negative 1 end exponent space equals space open square brackets table row cell 6 minus 12 plus 9 end cell cell space space space space space minus 5 plus 6 plus 0 end cell cell space space space space space space 5 minus 6 plus 0 end cell row cell negative 5 plus 6 plus 0 end cell cell space space space space 6 minus 12 plus 9 end cell cell space space space space minus 5 plus 6 plus 0 end cell row cell 5 minus 6 plus 0 end cell cell space space minus 5 plus 6 plus 0 end cell cell space space space space space 6 minus 12 plus 9 end cell end table close square brackets rightwards double arrow space space space space space space 4 straight A to the power of negative 1 end exponent space equals space open square brackets table row cell 6 minus 12 plus 9 end cell cell space space space space minus 5 plus 6 plus 0 end cell cell space space space space space space 5 minus 6 plus 0 end cell row cell negative 5 plus 6 plus 0 end cell cell space space space space space 6 minus 12 plus 9 end cell cell space space space minus 5 plus 6 plus 0 end cell row cell 5 minus 6 plus 0 end cell cell space minus 5 plus 6 plus 0 end cell cell space space space space space space 6 minus 12 plus 9 end cell end table close square brackets rightwards double arrow space space space space space space 4 straight A to the power of negative 1 end exponent space equals space open square brackets table row 3 cell space space space space 1 end cell cell space space space minus 1 end cell row 1 cell space space space space 3 end cell cell space space space space 1 end cell row cell negative 1 end cell cell space space space 1 end cell cell space space space 3 end cell end table close square brackets rightwards double arrow space space space space space straight A to the power of negative 1 end exponent space equals space 1 fourth open square brackets table row 3 cell space space space 1 end cell cell space space space minus 1 end cell row 1 cell space space space 3 end cell cell space space space space 1 end cell row cell negative 1 end cell cell space space 1 end cell cell space space space 3 end cell end table close square brackets.

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