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Determinants

Question
CBSEENMA12034584
Wired Faculty App

For the matrix straight A space equals space open square brackets table row 3 cell space space space 2 end cell row 1 cell space space space 1 end cell end table close square brackets comma find the numbers a and b such that A2 + aA + bI = O. Hence find A–1.

Solution
straight A space equals space open square brackets table row 3 cell space space space space 2 end cell row 1 cell space space space 1 end cell end table close square brackets
therefore space space space straight A squared space equals space open square brackets table row 3 cell space space space 2 end cell row 1 cell space space space 1 end cell end table close square brackets space space open square brackets table row 3 cell space space 2 end cell row 1 cell space space 1 end cell end table close square brackets space equals space open square brackets table row cell 9 plus 2 end cell cell space space space space 6 plus 2 end cell row cell 3 plus 1 end cell cell space space space 2 plus 1 end cell end table close square brackets space equals space open square brackets table row 11 cell space space space 8 end cell row 4 cell space space space 3 end cell end table close square brackets Now comma space straight A squared plus straight a space straight A space plus space straight b space straight I space equals space straight O rightwards double arrow space space space space open square brackets table row 11 cell space space space 8 end cell row 4 cell space space space 3 end cell end table close square brackets space plus space straight a space open square brackets table row 3 cell space space space 2 end cell row 1 cell space space 1 end cell end table close square brackets space plus space straight b space open square brackets table row 1 cell space 0 end cell row 0 cell space 1 end cell end table close square brackets space equals space open square brackets table row 0 cell space space 0 end cell row 0 cell space space 0 end cell end table close square brackets rightwards double arrow space space space space open square brackets table row 11 8 row 4 3 end table close square brackets space plus open square brackets table row cell 3 straight a end cell cell space space space 2 straight a end cell row straight a cell space space straight a end cell end table close square brackets space plus space open square brackets table row straight b cell space space 0 end cell row 0 cell space space space straight b space end cell end table close square brackets space equals space open square brackets table row 0 cell space space space 0 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow space space space space space space open square brackets table row cell 11 plus 3 straight a plus straight b end cell cell space space space space space space 8 plus 2 straight a end cell row cell 4 plus straight a end cell cell space space space space space 3 plus straight a plus straight b end cell end table close square brackets space equals open square brackets table row 0 cell space space space 0 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow space space space space space 4 plus straight a space equals space 0 comma space space space space 3 plus straight a plus 5 space equals space 0 space space space space rightwards double arrow space space space space straight a space space equals negative 4 comma space space 3 minus 4 plus straight b space equals space 0 therefore space space space space space space space space space space space space space space straight a space equals space minus 4 comma space space space space space straight b space equals space 1 therefore space space space space space space straight A squared minus 4 straight A plus straight I space equals space straight O space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight I space equals space minus straight A squared plus 4 straight A rightwards double arrow space space space space space space straight A to the power of negative 1 end exponent space equals space minus straight A plus 4 space straight I space equals space minus open square brackets table row 3 cell space space space 2 space end cell row 1 cell space 1 end cell end table close square brackets space plus space 4 space open square brackets table row 1 cell space space 0 end cell row 0 cell space space 1 end cell end table close square brackets rightwards double arrow space space space space space space straight A to the power of negative 1 end exponent space equals space open square brackets table row cell negative 3 end cell cell space space space space minus 2 end cell row cell negative 1 end cell cell space space space minus 1 end cell end table close square brackets space plus space open square brackets table row 4 cell space space space 0 end cell row 0 cell space space space 4 end cell end table close square brackets space space rightwards double arrow space space straight A to the power of negative 1 end exponent space equals space open square brackets table row cell negative 3 plus 4 end cell cell space space space minus 2 plus 0 end cell row cell negative 1 plus 0 end cell cell space space space minus 1 space plus space 4 end cell end table close square brackets rightwards double arrow space space space space straight A to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space space minus 2 end cell row cell negative 1 end cell cell space space space space space space 3 end cell end table close square brackets.

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