If $straight A space equals space open square brackets table row 1 cell space minus 1 end cell cell space space 1 end cell row 2 cell space 3 end cell cell space space 0 end cell row 18 cell space 2 end cell cell space space 10 end cell end table close square brackets comma$
then verify that A (adj. A) = O.

Solution

straight A space equals space open square brackets table row 1 cell space minus 1 end cell cell space 1 end cell row 2 cell space 3 end cell cell space 0 end cell row 18 cell space 2 end cell 10 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 rightwards double arrow space space space space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell negative 1 end cell 1 row 2 3 0 row 18 2 10 end table close vertical bar

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

open vertical bar table row 3 cell space space space space 0 end cell row 2 cell space space 10 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space 0 end cell row 18 cell space space 10 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space 3 end cell row 18 cell space space 2 end cell end table close vertical bar

i.e. 30 – 0, – (20 – 0), 4 – 54, i.e., 30, – 20, – 50 respectively
Co-factors of the elements of the second row of | A | are

negative space open vertical bar table row cell negative 1 end cell cell space space space space space 1 end cell row 2 cell space space space space 10 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space space 1 end cell row 18 cell space space 10 end cell end table close vertical bar comma space space minus open vertical bar table row 1 cell space space minus 1 end cell row 18 cell space space space space space 2 end cell end table close vertical bar

i.e. 12, – 8, – 20 respectively
Co-factors of the elements of the third row of | A | are

open vertical bar table row cell negative 1 end cell cell space space space 1 end cell row 3 cell space space 0 end cell end table close vertical bar comma space space minus open vertical bar table row 1 cell space space space space space space 1 end cell row 2 cell space space space space space 0 end cell end table close vertical bar comma space space open vertical bar table row 1 cell space space space space space minus 1 end cell row 2 cell space space space space space space space space 3 end cell end table close vertical bar comma space space straight i. straight e. comma space space minus 3 comma space space 2 comma space space space 5 space respectively

therefore space space space adj. space straight A space equals space open square brackets table row 30 cell space space minus 20 end cell cell space minus 50 end cell row 12 cell space minus 8 end cell cell space minus 20 end cell row cell negative 3 end cell cell space space 2 end cell cell space space space 5 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 30 cell space space space 12 end cell cell space space minus 3 end cell row cell negative 20 end cell cell space minus 8 end cell cell space space space space space 2 end cell row cell negative 50 end cell cell space space minus 20 end cell cell space space space space space 5 end cell end table close square brackets straight A left parenthesis adj. space straight A right parenthesis space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 2 cell space space 3 end cell cell space space space 0 end cell row 18 cell space space 2 end cell cell space 10 end cell end table close square brackets space open square brackets table row cell space 30 end cell cell space space space space 12 end cell cell space space space minus 3 end cell row cell negative 20 end cell cell space space minus 8 end cell cell space space space space space space 2 end cell row cell negative 50 end cell cell space space minus 20 end cell cell space space space space space 5 end cell end table close square brackets 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 30 plus 20 minus 50 end cell cell 12 plus 8 minus 20 end cell cell space space minus 3 minus 2 plus 5 end cell row cell 60 minus 60 plus 0 end cell cell 24 minus 24 plus 0 end cell cell space minus 6 plus 6 plus 0 end cell row cell 540 minus 40 minus 500 end cell cell space space space space space space 216 minus 16 minus 200 end cell cell space space space space minus 54 plus 4 plus 50 end cell end table close square brackets space equals open square brackets table row 0 cell space space 0 end cell cell space 0 end cell row 0 cell space 0 end cell cell space 0 end cell row 0 cell space 0 end cell cell space 0 end cell end table close square brackets therefore space space straight A left parenthesis Adj. space straight A right parenthesis space equals space straight O 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 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

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