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Determinants

Question
CBSEENMA12034573
Wired Faculty App

If straight A space equals space open square brackets table row 3 cell space space space space space 1 end cell row 7 cell space space space space space 5 end cell end table close square brackets comma find x and y find that A2 + xI = y A. Hence find A–1

Solution

 Here space straight A space equals space open square brackets table row 3 cell space space space space space 1 end cell row 7 cell space space space space space 5 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 space space 1 end cell row 7 cell space space space space 5 end cell end table close square brackets space open square brackets table row 3 cell space space space space 1 end cell row 7 cell space space space space 5 end cell end table close square brackets space equals space open square brackets table row cell 9 plus 7 end cell cell space space space space space 3 plus 5 end cell row cell 21 plus 35 end cell cell space space space space space 7 plus 25 end cell end table close square brackets space equals space open square brackets table row 16 cell space space 8 end cell row 56 cell space space 32 end cell end table close square brackets Now space straight A squared plus straight x space straight I space equals space straight y space straight A rightwards double arrow space space space space space open square brackets table row 16 cell space space space space 8 end cell row 56 cell space space space 32 end cell end table close square brackets space plus space straight x space open square brackets table row 1 cell space space space 0 end cell row 0 cell space space space 1 end cell end table close square brackets space equals space straight y open square brackets table row 3 cell space space space space 1 end cell row 7 cell space space space space 5 end cell end table close square brackets rightwards double arrow space space space space space open square brackets table row 16 cell space space space space space 8 end cell row 56 cell space space space space 32 end cell end table close square brackets space plus space open square brackets table row straight x cell space space space 0 end cell row 0 cell space space space straight x end cell end table close square brackets space equals space open square brackets table row cell 3 straight y end cell cell space space space straight y end cell row cell 7 straight y end cell cell space 5 straight y end cell end table close square brackets rightwards double arrow space space space space space open square brackets table row cell 16 plus straight x end cell cell space space space space space 8 end cell row 56 cell space space space 32 plus straight x end cell end table close square brackets space equals space open square brackets table row cell 3 straight y end cell cell space space space space straight y end cell row cell 7 straight y end cell cell space space space 5 straight y end cell end table close square brackets
 From the definition of equality of matrices,
                               space space space space space space space straight y space equals 8 space space space space space space space space space space space space space space space space space space 16 plus straight x space equals space 3 straight y therefore space space space space straight y space equals space 8 comma space space space space space space space space space space space space space space space space 16 plus straight x space equals 24 therefore space space space space straight y space equals space 8 comma 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 straight x space equals space 8 therefore space space space space straight A squared plus 8 thin space straight I space equals space 8 space straight A Post space multiplying space both space sides space by space straight A to the power of negative 1 end exponent comma space space space space space space space space space space space space space space straight A squared space straight A to the power of negative 1 end exponent plus space 8 space straight I thin space straight A to the power of negative 1 end exponent space equals space 8 space straight A thin space straight A to the power of negative 1 end exponent therefore space space space space space space space space space space space space space space space straight A plus space 8 space straight A to the power of negative 1 end exponent space space equals space 8 space straight I therefore space space space space space space space space space space space space space space space 8 space straight A to the power of negative 1 end exponent space space equals space 8 thin space straight I space minus space straight A 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 equals space 8 open square brackets table row 1 cell space space 0 end cell row cell space 0 end cell cell space space 1 end cell end table close square brackets minus space open square brackets table row 3 cell space space space 1 end cell row 7 cell space space space space 5 end cell end table close square brackets space equals space open square brackets table row 8 cell space space space 0 end cell row 0 cell space space space 8 end cell end table close square brackets plus open square brackets table row cell negative 3 end cell cell space space minus 1 end cell row cell negative 7 end cell cell space space space minus 5 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 equals space open square brackets table row cell 8 minus 3 end cell cell space space space 0 minus 1 end cell row cell 0 minus 7 end cell cell space space space 8 minus 5 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 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 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


                8 thin space straight A to the power of negative 1 end exponent space equals space open square brackets table row 5 cell space space space space space minus 1 end cell row cell negative 7 end cell cell space space space space space space 3 end cell end table close square brackets therefore space space space space space straight A to the power of negative 1 end exponent space equals space 1 over 8 open square brackets table row 5 cell space space space minus 1 end cell row cell negative 7 end cell cell space space space space space 3 end cell end table close square brackets.

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