find k so that A2 = k A - 2 1

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Question

$If space straight A equals open square brackets table row 3 cell negative 2 end cell row 4 cell negative 2 end cell end table close square brackets comma$ find k so that A² = k A - 2 1

Solution

space straight A equals open square brackets table row 3 cell negative 2 end cell row 4 cell negative 2 end cell end table close square brackets comma NOW space space space straight A squared equals straight k space straight A minus 21 rightwards double arrow open square brackets table row 3 cell negative 2 end cell row 4 cell negative 2 end cell end table close square brackets space open square brackets table row 3 cell negative 2 end cell row 4 cell negative 2 end cell end table close square brackets equals straight K space space open square brackets table row 3 cell negative 2 end cell row 4 cell negative 2 end cell end table close square brackets minus 2 open square brackets table row 1 0 row 0 1 end table close square brackets rightwards double arrow open square brackets table row cell 19 minus 8 end cell cell negative 6 plus 4 end cell row cell 12 minus 8 end cell cell negative 8 plus 4 end cell end table close square brackets equals open square brackets table row cell 3 straight k end cell cell space space minus 2 straight k end cell row cell 4 straight k end cell cell space space space minus 2 straight k end cell end table close square brackets plus open square brackets table row cell negative 2 end cell 0 row 0 cell negative 2 end cell end table close square brackets rightwards double arrow open square brackets table row 1 cell negative 2 end cell row 4 4 end table close square brackets equals open square brackets table row cell 3 straight k minus 2 end cell cell negative 2 straight k end cell row cell 4 straight k end cell cell negative 2 straight k minus 2 end cell end table close square brackets rightwards double arrow 1 equals 3 space straight k minus 2 comma negative 2 equals negative 2 straight k comma space 4 equals 4 straight k comma negative 4 equals negative 2 space straight k minus 2 rightwards double arrow space straight k equals 1

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Matrices

$If space straight A equals open square brackets table row 3 cell negative 2 end cell row 4 cell negative 2 end cell end table close square brackets comma$ find k so that A² = k A - 2 1

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