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Determinants

Question
CBSEENMA12034669
Wired Faculty App

For the matrix straight A space equals space open square brackets table row 1 cell space space 1 end cell cell space space space space space 1 end cell row 1 cell space space space 2 end cell cell space minus 3 end cell row 2 cell negative 1 end cell cell space space space space 3 end cell end table close square brackets comma show that A3 + 6A2 + 5A + 11 I = 0. Hence, find A–1.

Solution
straight A equals open square brackets table row 1 cell space space space 1 end cell cell space space space space 1 end cell row 1 cell space space space 2 end cell cell space minus 3 end cell row 2 cell space space space 1 space end cell cell space space space 3 end cell end table close square brackets
straight A squared space equals space open square brackets table row 1 cell space space space space space space 1 end cell cell space space space space space 1 end cell row 1 cell space space space space space space 2 end cell cell space space minus 3 end cell row 2 cell space minus 1 end cell cell space space space space space 3 end cell end table close square brackets space open square brackets table row 1 cell space space space 1 end cell cell space space space space 1 end cell row 1 cell space space space space 2 end cell cell space minus 3 end cell row 2 cell negative 1 end cell cell space space space space 3 end cell end table close square brackets space space space space space equals space open square brackets table row cell 1 plus 1 plus 2 end cell cell space space space space 1 plus 2 minus 1 end cell cell space space space space 1 minus 3 plus 3 end cell row cell 1 plus 2 minus 6 end cell cell space space 1 plus 4 plus 3 end cell cell space 1 minus 6 minus 9 end cell row cell 2 minus 1 plus 6 end cell cell space space 2 minus 2 minus 3 end cell cell space space 2 plus 3 plus 9 end cell end table close square brackets space equals space open square brackets table row cell space 4 end cell cell space space space 2 end cell cell space space space 1 end cell row cell negative 3 end cell cell space space space space 8 end cell cell space space 14 end cell row 7 cell space minus 3 end cell cell space space 14 end cell end table close square brackets straight A cubed space equals space open square brackets table row cell space space space 4 end cell cell space space space space 2 end cell cell space space space space space space 1 end cell row cell negative 3 end cell cell space space space space space 8 end cell cell space space minus 14 end cell row cell space space 7 end cell cell space space minus 3 end cell cell space space space space space 14 end cell end table close square brackets space open square brackets table row 1 cell space space space space space 1 end cell cell space space space 1 end cell row 1 cell space space space space 2 end cell cell space space minus 3 end cell row 2 cell space space minus 1 end cell cell space space space 3 end cell end table close square brackets space space space space space equals open square brackets table row cell 4 plus 2 plus 2 end cell cell space space space space space space space 4 plus 4 minus 1 end cell cell space space space space space 4 minus 6 plus 3 end cell row cell negative 3 plus 8 minus 28 end cell cell space space space space space minus 3 plus 16 plus 14 end cell cell space space space space minus 3 minus 24 minus 42 end cell row cell 7 minus 3 plus 28 end cell cell space space space space space 7 minus 6 minus 14 end cell cell space space space space space space space 7 plus 9 plus 42 end cell end table close square brackets space equals space open square brackets table row 8 cell space space 7 end cell cell space space 1 end cell row cell negative 23 end cell cell space space 27 end cell cell negative 69 end cell row 32 cell negative 13 end cell cell space space space 58 end cell end table close square brackets
Consider A3 + 6A2 + 5A + 11 I
equals space open square brackets table row 8 cell space space space 7 end cell cell space space space space 1 end cell row cell negative 23 end cell cell space space space space 27 end cell cell space space minus 69 end cell row 32 cell space minus 13 end cell cell space space space space 58 end cell end table close square brackets space minus space 6 open square brackets table row 4 cell space space space 2 end cell cell space space 1 end cell row cell negative 3 end cell cell space space space 8 end cell cell negative 14 end cell row 7 cell space minus 3 end cell cell space space 14 end cell end table close square brackets space plus space 5 space open square brackets table row 1 cell space 1 end cell cell space 1 end cell row 1 cell space space 2 end cell cell negative 3 end cell row 2 cell space minus 1 end cell cell space space space 3 end cell end table close square brackets plus space 11 open square brackets table row 1 cell space 0 end cell cell space 0 end cell row 0 cell space 1 end cell cell space space 0 end cell row 0 cell space 0 end cell cell space space 1 end cell end table close square brackets equals space open square brackets table row 8 cell space space 7 end cell cell space space 1 end cell row cell negative 23 end cell cell space space 27 end cell cell negative 69 end cell row 32 cell negative 13 end cell cell space space 58 end cell end table close square brackets plus space open square brackets table row cell negative 24 end cell cell space space space minus 12 end cell cell space space minus 6 end cell row 18 cell space minus 48 end cell cell space space space space 84 end cell row cell negative 42 end cell cell space 18 end cell cell negative 84 end cell end table close square brackets space plus space open square brackets table row 5 cell space space 5 end cell 5 row 5 cell space 10 end cell cell negative 15 end cell row 10 cell negative 5 end cell 15 end table close square brackets space plus space open square brackets table row 11 cell space 0 end cell 0 row 0 11 0 row 0 0 11 end table close square brackets space equals space open square brackets table row cell 8 minus 24 plus 5 plus 11 end cell cell space space space space space 7 minus 12 plus 5 plus 0 end cell cell space space space space space space 1 minus 6 plus 5 plus 0 end cell row cell negative 23 plus 18 plus 5 plus 0 end cell cell space space space space space space space 27 minus 48 plus 10 plus 11 end cell cell space space space minus 69 plus 84 minus 15 plus 0 end cell row cell 32 minus 42 plus 10 plus 0 end cell cell negative 13 plus 18 minus 5 plus 0 end cell cell space space space space space space space 58 minus 84 plus 15 plus 11 end cell end table close square brackets space equals space open square brackets table row 0 cell space 0 end cell cell 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 therefore space space space straight A cubed minus 6 straight A squared plus 5 straight A plus 11 space straight I space equals space straight O rightwards double arrow space space space space space space space 11 space straight I space equals space minus straight A cubed plus 6 straight A squared minus 5 straight A rightwards double arrow space space space space space space space 11 space straight A to the power of negative 1 end exponent space space equals negative straight A squared plus 6 straight A minus 5 thin 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 equals negative open square brackets table row 4 cell space space space space 2 end cell cell space space space space 1 end cell row cell negative 3 end cell cell space space space 8 end cell cell space space minus 14 end cell row 7 cell space minus 3 end cell cell space minus 14 end cell end table close square brackets plus space 6 space open square brackets table row 1 cell space space space space space space space 1 end cell cell space space space space space 1 end cell row 1 cell space space space space space space space 2 end cell cell space space minus 3 end cell row 2 cell space space minus 1 end cell cell space space space space 3 end cell end table close square brackets minus 5 open square brackets table row 1 cell space space space 0 end cell cell space 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 space space 1 end cell end table close square brackets therefore space space space space space space space 11 space straight A to the power of negative 1 end exponent space equals space open square brackets table row cell negative 4 end cell cell space space space minus 2 end cell cell space space minus 1 end cell row cell space space 3 end cell cell space space minus 8 end cell cell space space space space space 14 end cell row cell negative 7 end cell cell space space space 3 end cell cell negative 14 end cell end table close square brackets plus open square brackets table row 6 cell space space 6 end cell cell space space space 6 end cell row 6 cell space 12 end cell cell negative 18 end cell row 12 cell negative 6 end cell cell space 18 end cell end table close square brackets space plus open square brackets table row cell negative 5 end cell cell space space space space 0 end cell cell space space space space 0 end cell row 0 cell space minus 5 end cell cell space space space space 0 end cell row 0 cell space space 0 end cell cell space space minus 5 end cell end table close square brackets space space space space space space space space space space space space space
rightwards double arrow space space 11 space straight A to the power of negative 1 end exponent space equals space open square brackets table row cell negative 4 plus 6 minus 5 end cell cell space space space space minus 2 plus 6 plus 0 end cell cell space space space space space space minus 1 plus 6 plus 0 end cell row cell 3 plus 6 plus 0 end cell cell space minus 8 plus 12 minus 5 end cell cell space space 14 minus 18 plus 0 end cell row cell negative 7 plus 12 plus 0 end cell cell space 3 minus 6 plus 0 end cell cell space space minus 14 plus 18 minus 5 end cell end table close square brackets rightwards double arrow space space space space space 11 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 4 end cell cell space space space 5 end cell row 9 cell space minus 1 end cell cell space minus 4 end cell row 5 cell negative 3 end cell cell space minus 1 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 over 11 open square brackets table row cell negative 3 end cell cell space space space 4 end cell cell space space space space 5 end cell row 9 cell negative 1 end cell cell space minus 4 end cell row 5 cell negative 3 end cell cell space minus 1 end cell end table close square brackets

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