If $straight A space equals space open square brackets table row 1 cell space space 2 end cell cell space space 2 end cell row 2 cell space space 1 end cell cell space space 2 end cell row 2 cell space space 2 end cell cell space space 1 end cell end table close square brackets comma space$ Find A^–1 and hence prove that A² – 4A – 5 I = O.

Solution

straight A space equals space open square brackets table row 1 cell space space space 2 end cell cell space space space 2 end cell row 2 cell space space space 1 end cell cell space space space 2 end cell row 2 cell space space space 2 end cell cell space space 1 end cell end table close square brackets therefore space space space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space 2 end cell cell space space space 2 end cell row 2 cell space space space 1 end cell cell space space space 2 end cell row 2 cell space space space 2 end cell cell space space 1 end cell end table close vertical bar space space equals space 1 open vertical bar table row 1 cell space space space 2 end cell row 2 cell space space space 1 end cell end table close vertical bar minus 2 open vertical bar table row 2 cell space space space space 2 end cell row 2 cell space space space space 1 end cell end table close vertical bar plus 2 open vertical bar table row 2 cell space space space space 1 end cell row 2 cell space space space space space 2 end cell end table close vertical bar space space space space space space space space space space space space space space space space space equals 1 left parenthesis 1 minus 4 right parenthesis minus 2 left parenthesis 2 minus 4 right parenthesis plus 2 left parenthesis 4 minus 2 right parenthesis space equals space minus 3 plus 4 plus 4 space equals 5 therefore space space space space space straight A to the power of negative 1 end exponent space exists

Co-factors of the elements of the first row of | A | are

open vertical bar table row 1 cell space space 2 end cell row 2 cell space space 1 end cell end table close vertical bar comma space space space minus open vertical bar table row 2 cell space space space space 2 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space space 1 end cell row 2 cell space space space 2 end cell end table close vertical bar

i.e. – 3, 2, 2 respectively
Co-factors of the elements of the second row of | A | are

negative open vertical bar table row 2 cell space space space 2 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space space space open vertical bar table row 1 cell space space space 2 end cell row 2 cell space space 1 end cell end table close vertical bar comma space space space minus open vertical bar table row 1 cell space space 2 end cell row 2 cell space space space 2 end cell end table close vertical bar

i.e. 2, – 3, 2 respectively.
Co-factors of the elements of the third row of | A | are

open vertical bar table row 2 cell space space 2 end cell row 1 cell space space 2 end cell end table close vertical bar comma space space space minus open vertical bar table row 1 cell space space space 2 end cell row 2 cell space space space 2 end cell end table close vertical bar comma space space open vertical bar table row 1 cell space space space 2 end cell row 2 cell space space 1 end cell end table close vertical bar straight i. straight e. comma space space 2 comma space space 2 comma space – space 3 space respectively

therefore space space space adj. space straight A space equals space open square brackets table row cell negative 3 end cell cell space space space space space space 2 end cell cell space space space space space 2 end cell row 2 cell space space minus 3 end cell cell space space space space space 2 end cell row 2 cell space space space space space 2 end cell cell space space minus 3 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 3 end cell cell space space space space space space 2 end cell cell space space space space space 2 end cell row 2 cell space space minus 3 end cell cell space space space space space 2 end cell row 2 cell space space space space space 2 end cell cell space space minus 3 end cell end table close square brackets Now space straight A to the power of negative 1 end exponent space equals space fraction numerator adj. space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space 1 fifth open square brackets table row cell negative 3 end cell cell space space 2 end cell cell space space space space 2 end cell row cell space space 2 end cell cell negative 3 end cell cell space space space space 2 end cell row cell space space 2 end cell cell space space 2 end cell cell negative 3 end cell end table close square brackets. space therefore space space space space space space space 5 space straight A to the power of negative 1 end exponent space equals space open square brackets table row cell 1 minus 4 end cell cell space space space space 2 plus 0 end cell cell space space space space space 2 plus 0 end cell row cell 2 plus 0 end cell cell space space space 1 minus 4 end cell cell space space space 2 plus 0 end cell row cell 2 plus 0 end cell cell space space space 2 plus 0 end cell cell space space space 1 minus 4 end cell end table close square brackets rightwards double arrow space space space space space space 5 space straight A to the power of negative 1 end exponent space equals space space open square brackets table row 1 cell space space space 2 end cell cell space space 2 end cell row 2 cell space space 1 end cell cell space space 2 end cell row 2 cell space space 2 end cell cell space space 1 end cell end table close square brackets plus space open square brackets table row cell negative 4 end cell cell space space space 0 end cell cell space space space space space space 0 end cell row 0 cell space minus 4 end cell cell space space space space space 0 end cell row 0 cell space space 0 end cell cell space minus 4 end cell end table close square brackets rightwards double arrow space space space space space 5 space straight A to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space 2 end cell cell space space 2 end cell row 2 cell space 1 end cell cell space space 2 end cell row 2 cell space space 2 end cell cell space 1 end cell end table close square brackets space minus space 4 open square brackets table row 1 cell space 0 end cell cell space 0 end cell row 0 cell space 1 end cell cell space 0 end cell row 0 cell space space 0 end cell cell space 1 end cell end table close square brackets rightwards double arrow space space space space 5 space straight A to the power of negative 1 end exponent space equals space straight A space minus space 4 space straight I rightwards double arrow space space space space 5 space straight A space straight A to the power of negative 1 end exponent space equals space AA space minus space 4 straight A space straight I rightwards double arrow space space space space space space space 5 space straight I space equals space straight A squared minus 4 space straight A rightwards double arrow space space space space space space straight A squared minus 4 straight A space minus space 5 space straight I space equals space 0.

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