Question

$If space straight A space equals space open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets space and space straight A cubed minus 6 straight A squared plus 7 straight A space plus kI subscript 3 space equals space 0 space find space straight k.$

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Solution

straight A space equals open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets straight A squared space equals space AA space equals space open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets equals open square brackets table row 5 0 8 row 2 4 5 row 8 0 13 end table close square brackets straight A cubed space equals space straight A squared. straight A space equals space open square brackets table row 5 0 8 row 2 4 5 row 8 0 13 end table close square brackets open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets space equals open square brackets table row 21 0 34 row 12 8 23 row 34 0 55 end table close square brackets therefore comma space straight A cubed minus 6 straight A squared plus 7 straight A plus KI subscript 3 space equals space 0 rightwards double arrow space open square brackets table row 21 0 34 row 12 8 23 row 34 0 55 end table close square brackets minus 6 open square brackets table row 5 0 8 row 2 4 5 row 8 0 13 end table close square brackets plus 7 open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets plus straight k open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets space equals 0 rightwards double arrow space straight k equals 2

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