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Determinants

Question
CBSEENMA12034538
Wired Faculty App

Verify that A (adj A) = (adj A) = | A | I
where straight A space equals space open square brackets table row 2 cell space space space 1 end cell cell space space space space space 5 end cell row 3 cell space minus 2 end cell cell space space space minus 4 end cell row cell negative 3 end cell cell space 1 end cell cell space space minus 2 end cell end table close square brackets

Solution
straight A space equals space open square brackets table row 2 cell space space space 1 end cell cell space space space space space space space 5 end cell row 3 cell negative 2 end cell cell space space space minus 4 end cell row cell negative 3 end cell cell space 1 end cell cell space space space space minus 2 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 cell space space space 2 end cell cell space space space space space 1 end cell cell space space space space space 5 end cell row cell space space space 3 end cell cell space minus 2 end cell cell space space space minus 4 space end cell row cell negative 3 end cell cell space space space 1 end cell cell space minus 2 end cell end table close vertical bar space equals space 2 space open vertical bar table row cell negative 2 end cell cell space space minus 4 end cell row 1 cell space space minus 2 end cell end table close vertical bar space minus space 1 open vertical bar table row cell space space space 3 end cell cell space space space minus 4 end cell row cell negative 3 end cell cell space space space space minus 2 end cell end table close vertical bar space plus space 5 space open vertical bar table row cell space space 3 end cell cell space minus 2 end cell row cell negative 3 end cell cell space space space space 1 end cell end table close vertical bar space space space space space space space space space space space space space space space space equals space 2 left parenthesis 4 plus 4 right parenthesis space minus 1 left parenthesis negative 6 minus 12 right parenthesis space plus 5 space left parenthesis 3 minus 6 right parenthesis space equals space 16 plus 18 minus 15 space equals space 19 space space space space space straight A subscript 11 equals space open vertical bar table row cell negative 2 end cell cell space space space minus 4 end cell row 1 cell space space space minus 2 end cell end table close vertical bar space equals space 4 plus 4 space equals space 8 space space space space space straight A subscript 12 space equals space minus open vertical bar table row 3 cell space space space space minus 4 end cell row cell negative 3 end cell cell space space space minus 2 end cell end table close vertical bar space equals space minus left parenthesis negative 6 minus 12 right parenthesis space equals space 18 space space space space space straight A subscript 13 space equals space open vertical bar table row cell space space 3 end cell cell space space space space minus 2 end cell row cell negative 3 end cell cell space space space space space space 1 end cell end table close vertical bar space equals space 3 minus 6 space equals negative 3 space space space space straight A subscript 21 space equals space minus open vertical bar table row 1 cell space space space space space space space 5 end cell row 1 cell space space space minus 2 end cell end table close vertical bar space equals space minus left parenthesis negative 2 minus 5 right parenthesis space equals space 7 space space space space space straight A subscript 22 space equals space open vertical bar table row 2 cell space space space space space space space 5 end cell row 3 cell space space space space space minus 2 end cell end table close vertical bar space equals space minus 4 plus 15 space equals space 11 space space space space space straight A subscript 23 space equals space minus open vertical bar table row 2 cell space space space space space space 1 end cell row cell negative 3 end cell cell space space space space space 1 end cell end table close vertical bar space equals space minus left parenthesis 2 plus 3 right parenthesis space equals space minus 5 space space space space straight A subscript 31 space equals space open vertical bar table row 1 cell space space space 5 end cell row cell negative 2 end cell cell space space minus 4 end cell end table close vertical bar space equals space minus 4 plus 10 space equals space 6
    straight A subscript 32 space equals space minus open vertical bar table row 2 cell space space space space space space 5 end cell row 3 cell space space minus 4 end cell end table close vertical bar space equals space minus left parenthesis negative 8 minus 15 right parenthesis space equals space 23 straight A subscript 33 space equals space open vertical bar table row 2 cell space space space space space space space 1 end cell row 3 cell space space space minus 2 end cell end table close vertical bar space equals space minus 4 minus 3 space equals space minus 7

therefore space space space space adj space straight A space equals space open square brackets table row 8 cell space space 18 end cell cell space minus 3 end cell row 7 cell space 11 end cell cell space minus 5 end cell row 6 cell space 23 end cell cell space minus 7 end cell end table close square brackets space equals space open square brackets table row 8 cell space 7 end cell cell space 6 end cell row 18 cell space 11 end cell cell space 23 end cell row cell negative 3 end cell cell space minus 5 end cell cell space minus 7 end cell end table close square brackets straight A space left parenthesis adj space straight A right parenthesis space equals space open square brackets table row 2 cell space space 1 end cell cell space space space space space 5 end cell row 3 cell space minus 2 end cell cell space space minus 4 end cell row cell negative 3 end cell cell space space 1 end cell cell space space space space minus 2 end cell end table close square brackets space open square brackets table row 8 cell space 7 end cell cell space space 6 end cell row 18 cell space 11 end cell cell space space 23 end cell row cell negative 3 end cell cell space minus 5 end cell cell space space minus 7 end cell end table close square brackets space space space space space space space space space space space space space space space space equals space open square brackets table row cell 16 plus 18 minus 15 end cell cell space space space space space space 14 plus 11 minus 25 end cell cell space space space 12 plus 23 minus 35 end cell row cell 24 minus 36 plus 12 end cell cell space space space 21 minus 22 plus 20 end cell cell space space space 18 minus 46 plus 28 end cell row cell negative 24 plus 18 plus 6 end cell cell space space space minus 21 plus 11 plus 10 end cell cell space space space minus 18 plus 23 plus 14 end cell end table close square brackets space space space space space space space space space space space space space space space space equals space open square brackets table row 19 0 0 row 0 19 0 row 0 0 19 end table close square brackets space space equals 19 space open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets therefore space space space space straight A left parenthesis adj space straight A right parenthesis space equals space open vertical bar straight A close vertical bar space straight I space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis Again space left parenthesis adj space straight A right parenthesis thin space straight A space equals space open square brackets table row 8 cell space 7 end cell cell space space 6 end cell row 18 cell space space 11 end cell cell space space 23 end cell row cell negative 3 end cell cell space minus 5 end cell cell negative 7 end cell end table close square brackets space space open square brackets table row 2 cell space space 1 end cell cell space space space 5 end cell row 3 cell space minus 2 end cell cell space space space minus 4 end cell row cell negative 3 end cell cell space 1 end cell cell space space minus 2 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open square brackets table row cell 16 plus 21 minus 18 end cell cell space space space 8 minus 14 plus 6 end cell cell space space space 40 minus 28 minus 12 end cell row cell 36 plus 33 minus 69 end cell cell space 18 minus 22 plus 23 end cell cell space 90 minus 44 minus 46 end cell row cell negative 6 minus 15 plus 21 end cell cell space minus 3 plus 10 minus 7 end cell cell space minus 15 plus 20 plus 14 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open square brackets table row 19 cell space 0 end cell 0 row 0 19 0 row 0 0 19 end table close square brackets space equals space 19 open square brackets table row 1 cell space 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 0 end cell cell space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
therefore space space space space left parenthesis adj space straight A right parenthesis thin space straight A space equals space open vertical bar straight A close vertical bar space straight I                                                     ...(2)
From (1) and (2), we get,
A (adj A) = (adj A) A = | A | I.

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