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Determinants

Question
CBSEENMA12034588
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For the matrix straight A space equals space open square brackets table row 3 cell space 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 so that A2 + xI = yA. Hence find A–1 .

Solution

Here straight A space equals space open square brackets table row 3 cell space space space 1 end cell row 7 cell space space space 5 end cell end table close square brackets
         straight A squared space equals space open square brackets table row 3 cell space space space 1 end cell row 7 cell space space space 5 end cell end table close square brackets space open square brackets table row 3 cell space space space 1 end cell row 7 cell 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 3 plus 5 end cell row cell 21 plus 35 end cell cell 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 space space rightwards double arrow space space 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 space plus space straight x space open square brackets table row 1 cell space 0 end cell row 0 cell space space 1 end cell end table close square brackets space equals space straight y open square brackets table row 3 cell space space 1 end cell row 7 cell space space 5 end cell end table close square brackets rightwards double arrow space space space space open square brackets table row 16 cell space space space 8 end cell row 56 cell 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 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 space space rightwards double arrow space space space open square brackets table row cell 16 plus straight x end cell cell space space space space 8 end cell row 56 cell 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 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 16 plus straight x space equals space 3 straight y comma space space 8 space equals straight y space space space space space space space rightwards double arrow space space 16 plus straight x space equals space 24 comma space space straight y space equals space 8 space space space space space space rightwards double arrow space space space straight x space equals space 8 comma space space straight y space equals space 8 therefore space space space space we space have space straight A squared plus space 8 space straight I space equals space 8 space straight A rightwards double arrow space space space space space space 8 space straight I space equals space 8 space straight A space minus space straight A squared space space space space rightwards double arrow space space space 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 space minus space straight A squared straight A to the power of negative 1 end exponent space space rightwards double arrow space space space 8 space straight A to the power of negative 1 end exponent space equals space 8 space straight I space minus space straight A space rightwards double arrow space space space 8 space straight A to the power of negative 1 end exponent space equals space 8 open square brackets table row 1 cell space space space 0 end cell row 0 cell space space 1 end cell end table close square brackets space minus space open square brackets table row 3 cell space space 1 end cell row 7 cell space space 5 end cell end table close square brackets space equals space open square brackets table row 8 cell space space 0 end cell row 0 cell space space 8 end cell end table close square brackets space plus space 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 minus 5 end cell end table close square brackets space equals space 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 therefore 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 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

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