Question

$If space straight A equals space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets space space is space such space that space straight A squared space equals space straight I comma space then left parenthesis straight A right parenthesis space straight l space plus space straight alpha squared space plus space βγ space equals space 0 space left parenthesis straight B right parenthesis space straight l – straight alpha squared plus space βγ space equals space 0 space space left parenthesis straight C right parenthesis space 1 space – straight alpha squared space – space βγ space equals space 0 space left parenthesis straight D right parenthesis space straight l space plus straight alpha squared – space βγ space equals space 0$

Solution

space straight A equals space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets now space straight A squared space equals 1 space rightwards double arrow space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets space equals space open square brackets table row 1 cell space space space 1 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow space open square brackets table row cell straight alpha to the power of 2 space space end exponent plus space βγ end cell cell γβ minus αβ end cell row cell γα minus γα end cell cell βγ plus straight alpha squared end cell end table close square brackets equals space open square brackets table row 1 cell space space space 1 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow open square brackets table row cell straight alpha to the power of 2 space space end exponent plus space βγ end cell 0 row 0 cell straight alpha to the power of 2 space space end exponent plus space βγ end cell end table close square brackets space equals space open square brackets table row 1 cell space space space 1 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow space by space definition space of space equality space of space matrices comma straight alpha to the power of 2 space space end exponent plus space βγ space equals 1 space space or space 1 minus straight alpha to the power of 2 space space end exponent plus space βγ space equals 0 therefore space space left parenthesis straight C right parenthesis space is space correct space answer. space

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Matrices

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