If $straight A space equals space open square brackets table row 1 cell space space 3 end cell cell space space space 3 end cell row 1 cell space space 4 end cell cell space space 3 end cell row 1 cell space 3 end cell cell space space space 4 end cell end table close square brackets comma$ then verify that A adj A = | A | I. Also find A .

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Solution

straight A space equals space open square brackets table row 1 cell space 3 end cell cell space space 3 end cell row 1 cell space 4 end cell cell space 3 end cell row 1 cell space 3 end cell cell space 4 end cell end table close square brackets

open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space 3 end cell cell space 3 end cell row 1 cell space 4 end cell cell space 3 end cell row 1 cell space 3 end cell cell space 4 end cell end table close vertical bar space equals space open vertical bar table row 4 cell space space 3 end cell row 3 cell space space 4 end cell end table close vertical bar minus 3 space open vertical bar table row 1 cell space 3 end cell row 1 cell space 4 end cell end table close vertical bar space plus space 3 space open vertical bar table row 1 cell space space 4 end cell row 1 cell space space 3 end cell end table close vertical bar space space space space space space space equals space 1 left parenthesis 16 minus 9 right parenthesis space minus 3 space left parenthesis 4 minus 3 right parenthesis space plus space 3 left parenthesis 3 minus 4 right parenthesis space equals space 7 minus 3 minus 3 space equals space 1 not equal to 0 therefore space space space straight A to the power of negative 1 end exponent space exists. space space space space space space straight A subscript 11 space equals space open vertical bar table row 4 cell space space 3 end cell row 3 cell space space 4 end cell end table close vertical bar space equals space 16 minus 9 space equals space 7 comma space space space space space straight A subscript 12 space equals space minus open vertical bar table row 1 cell space space space 3 end cell row 1 cell space space 4 end cell end table close vertical bar space equals space minus left parenthesis 4 minus 3 right parenthesis space equals space minus 1 space space space space space space straight A subscript 13 space equals space open vertical bar table row 1 cell space space space 4 end cell row 1 cell space space space 3 end cell end table close vertical bar space equals space 3 minus 4 space equals space minus 1 comma space space space straight A subscript 21 space equals space minus open vertical bar table row 3 cell space space space 3 end cell row 3 cell space space space 4 end cell end table close vertical bar space equals space minus left parenthesis 12 minus 9 right parenthesis space equals space minus 3 space space space space space space straight A subscript 22 space equals space open vertical bar table row 1 cell space space 3 end cell row 1 cell space space 4 end cell end table close vertical bar space equals space 4 minus 3 space equals space 1 comma space space space space straight A subscript 23 space equals space minus open vertical bar table row 1 cell space space space 3 end cell row 1 cell space space space 3 end cell end table close vertical bar space equals space minus left parenthesis 3 minus 3 right parenthesis space equals space 0 space space space space space space straight A subscript 31 space equals space open vertical bar table row 3 cell space space space 3 end cell row 4 cell space space space 3 end cell end table close vertical bar space equals space 9 minus 12 space equals space minus 3 comma space space space straight A subscript 32 space equals space minus open vertical bar table row 1 cell space space space 3 end cell row 1 cell space space space 3 end cell end table close vertical bar space equals space minus left parenthesis 3 minus 3 right parenthesis space equals space 0 space space space space space straight A subscript 33 space equals space open vertical bar table row 1 cell space space space 3 end cell row 1 cell space space space 4 end cell end table close vertical bar space equals space 4 minus 3 space equals space 1 space space space space space space adj. space straight A space equals space open square brackets table row cell straight A subscript 11 end cell cell space space straight A subscript 12 end cell cell space space space space straight A subscript 13 end cell row cell straight A subscript 12 end cell cell straight A subscript 22 end cell cell space space straight A subscript 23 end cell row cell straight A subscript 31 end cell cell straight A subscript 32 end cell cell space space straight A subscript 33 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 7 cell space space space minus 1 end cell cell space space space minus 1 end cell row cell negative 3 end cell cell space space space space space 1 end cell cell space space space space space space 0 end cell row cell negative 3 end cell cell space space space space 0 end cell cell space space space space space 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 7 cell space space minus 3 end cell cell space space minus 3 end cell row cell negative 1 end cell 1 cell space space space 0 end cell row cell negative 1 end cell 0 cell space space space 1 end cell end table close square brackets Now space space space space straight A left parenthesis adj space straight A right parenthesis space equals space open square brackets table row 1 cell space space 3 end cell cell space space space 3 end cell row 1 cell space space 4 end cell cell space space 3 end cell row 1 cell space space 3 end cell cell space space space 4 end cell end table close square brackets space space open square brackets table row cell space space space 7 end cell cell space space minus 3 end cell cell space space space minus 3 end cell row cell negative 1 end cell cell space space space space space 1 end cell cell space space space space space space 0 end cell row cell negative 1 end cell cell space space space space 0 end cell cell space space space space space 1 end cell end table close square brackets

equals space open square brackets table row cell 7 minus 3 minus 3 end cell cell space space space minus 3 plus 3 plus 0 end cell cell space space minus 3 plus 0 plus 3 end cell row cell 7 minus 4 minus 3 end cell cell space space minus 3 plus 4 plus 0 end cell cell space space minus 3 plus 0 plus 3 end cell row cell 7 minus 3 minus 4 end cell cell space minus 3 plus 3 plus 0 end cell cell space space space minus 3 plus 0 plus 4 end cell end table close square brackets space equals space open square brackets table row 1 cell space space 0 end cell cell space space space 0 end cell row 0 cell space space 1 end cell cell space space space 0 end cell row 0 cell space space 0 end cell cell space space space 1 end cell end table close square brackets space equals space left parenthesis 1 right parenthesis space open square brackets table row 1 cell space space 0 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 0 cell space space 0 end cell cell space space 1 end cell end table close square brackets space equals space open vertical bar straight A close vertical bar space straight I therefore space space space space space straight A left parenthesis adj space straight A right parenthesis space space equals open vertical bar straight A close vertical bar space space straight I Now space space space space space space space space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator 1 over denominator open vertical bar straight A close vertical bar end fraction space adj space straight A space equals space 1 over 1 open square brackets table row 7 cell space space space minus 3 end cell cell space space minus 3 end cell row cell negative 1 end cell cell space space space space space 1 end cell cell space space space space space 0 end cell row cell negative 1 end cell cell space space space space space space space 0 space end cell cell space space space space 1 end cell end table close square brackets space equals space open square brackets table row cell space space 7 end cell cell space space space minus 3 end cell cell space space minus 3 end cell row cell negative 1 end cell cell space space space space space space 1 end cell cell space space space space space 0 end cell row cell negative 1 end cell cell space space space space space space 0 end cell cell space space space space space 1 end cell end table close square brackets

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