Question

$If space straight A space equals space open square brackets table row 1 2 2 row 2 1 2 row 2 2 1 end table close square brackets comma space then space show space that space straight A squared minus 4 straight A minus 5 straight I space equals 0 comma space and space hence space find space straight A to the power of negative 1 end exponent space space space space space space space space space space space space space space space space space space space space space space space space$

Solution

straight A space equals space open square brackets table row 1 2 2 row 2 1 2 row 2 2 1 end table close square brackets straight A squared equals open square brackets table row 1 2 2 row 2 1 2 row 2 2 1 end table close square brackets space open square brackets table row 1 2 2 row 2 1 2 row 2 2 1 end table close square brackets space space space equals space open square brackets table row cell 1 cross times 1 plus 2 cross times 2 plus 2 cross times 2 end cell cell space space 1 cross times 2 plus 2 cross times 1 plus 2 cross times 2 end cell cell space space space space space 1 cross times 2 plus 2 cross times 2 plus 2 cross times 1 end cell row cell 2 cross times 1 plus 1 cross times 2 plus 2 cross times 2 end cell cell space 2 cross times 2 plus 1 cross times 1 plus 2 cross times 2 end cell cell space space space space 2 cross times 2 plus 1 cross times 2 plus 2 cross times 1 end cell row cell 2 cross times 1 plus 2 cross times 2 plus 1 cross times 2 end cell cell space 2 cross times 2 plus 2 cross times 1 plus 1 cross times 2 end cell cell space space space space 2 cross times 2 plus 2 cross times 2 plus 1 cross times 1 end cell end table close square brackets space equals space open square brackets table row cell 1 plus 4 plus 4 end cell cell space space space 2 plus 2 plus 4 end cell cell space space 2 plus 4 plus 2 end cell row cell 2 plus 2 plus 4 end cell cell space space 4 plus 1 plus 4 end cell cell space space 4 plus 2 plus 2 end cell row cell 2 plus 4 plus 2 end cell cell space 4 plus 2 plus 2 end cell cell space 4 plus 4 plus 1 end cell end table close square brackets equals space open square brackets table row 9 8 8 row 8 9 8 row 8 8 9 end table close square brackets

Conisder space straight A squared minus 4 straight A minus 5 straight I space equals space open square brackets table row 9 8 8 row 8 9 8 row 8 8 9 end table close square brackets space minus space 4 open square brackets table row 1 cell space 2 end cell cell space 2 end cell row 2 1 2 row 2 cell space 2 end cell 1 end table close square brackets space minus space 5 open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets space equals open square brackets table row 9 cell space 8 end cell cell space 8 end cell row 8 cell space 9 end cell cell space 8 end cell row 8 cell space 8 end cell cell space 9 end cell end table close square brackets space minus space open square brackets table row 4 cell space 8 end cell cell space 8 end cell row 8 cell space 4 end cell cell space 8 end cell row 8 cell space 8 end cell cell space 4 end cell end table close square brackets space minus space open square brackets table row 5 cell space 0 end cell cell space 0 end cell row 0 cell space 5 end cell cell space 0 end cell row 0 0 cell space 5 end cell end table close square brackets space equals space open square brackets table row cell 9 minus 9 end cell cell space 8 minus 8 end cell cell space space 8 minus 8 end cell row cell 8 minus 8 end cell cell space 9 minus 9 end cell cell space 8 minus 8 end cell row cell 8 minus 8 end cell cell space 8 minus 8 end cell cell space 9 minus 9 end cell end table close square brackets equals space open square brackets table row 0 0 0 row 0 0 0 row 0 0 0 end table close square brackets

Now straight A squared minus 4 straight A minus 5 straight I space equals space 0 straight A squared minus 4 straight A space equals space 5 straight I straight A squared straight A to the power of negative 1 end exponent space minus 4 AA to the power of negative 1 end exponent space equals space 5 IA to the power of negative 1 end exponent left parenthesis Postmultiply space by space straight A to the power of negative 1 end exponent right parenthesis straight A minus 4 straight I space equals space 5 straight A to the power of negative 1 end exponent open square brackets table row 1 cell space 2 end cell cell space 2 end cell row 2 1 cell space 2 end cell row 2 cell space 2 end cell cell space 1 end cell end table close square brackets space minus space open square brackets table row 4 cell space 0 end cell cell space 0 end cell row 0 cell space 4 end cell cell space 0 end cell row 0 cell space 0 end cell cell space 4 end cell end table close square brackets space equals space 5 straight A to the power of negative 1 end exponent open square brackets table row cell negative 3 end cell cell space space space 2 end cell cell space space space 2 end cell row 2 cell negative 3 end cell cell space space space 2 end cell row 2 cell space space 2 end cell cell negative 3 end cell end table close square brackets space equals space 5 straight A to the power of negative 1 end exponent

straight A to the power of negative 1 end exponent space equals space open square brackets table row cell fraction numerator negative 3 over denominator 5 end fraction end cell cell space 2 over 5 end cell cell space 2 over 5 end cell row cell 2 over 5 end cell cell fraction numerator negative 3 over denominator 5 end fraction end cell cell 2 over 5 end cell row cell 2 over 5 end cell cell 2 over 5 end cell cell fraction numerator negative 3 over denominator 5 end fraction end cell end table close square brackets

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Matrices

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