Question

$If space straight A equals space open square brackets table row cell 1 space space end cell cell space space space 0 end cell cell space space space 2 end cell row 0 cell space space 2 end cell cell space space space 1 end cell row 2 cell space 0 end cell cell space 3 end cell end table close square brackets space then space show space that space straight A cubed space minus 6 straight A squared space plus 7 straight A space plus space 2 space straight I equals space straight O$

Solution

H e r e space space straight A equals space open square brackets table row cell 1 space end cell 0 cell space space 2 end cell row cell 0 space space end cell 2 cell space 1 end cell row 2 cell space 0 end cell cell space 3 end cell end table close square brackets straight A squared space equals open square brackets table row cell 1 space end cell 0 cell space space 2 end cell row cell 0 space space end cell 2 cell space 1 end cell row 2 cell space 0 end cell cell space 3 end cell end table close square brackets space space open square brackets table row cell 1 space end cell 0 cell space space 2 end cell row cell 0 space space end cell 2 cell space 1 end cell row 2 cell space 0 end cell cell space 3 end cell end table close square brackets space space space space space space equals space space open square brackets table row cell 1 plus 0 plus 4 space space space end cell cell space space 0 plus 0 plus 0 end cell cell space space space 2 plus 0 plus 6 end cell row cell 0 plus 0 plus 2 end cell cell space space 0 plus 4 plus 0 end cell cell space space space 0 plus 2 plus 3 end cell row cell 2 plus 0 plus 6 end cell cell space space 0 plus 0 plus 0 end cell cell space space 4 plus 0 plus 9 end cell end table close square brackets space space equals space open square brackets table row 5 cell space space space space 0 end cell cell space space 8 end cell row 2 cell space space space 4 end cell cell space space 5 end cell row 8 cell space space space 0 end cell cell space space 13 end cell end table close square brackets straight A cubed space equals space space open square brackets table row cell 5 space space space end cell cell space space space 0 end cell cell space space space 8 end cell row 2 cell space space 4 end cell cell space space 5 end cell row 8 cell space space 0 end cell cell space space space 13 end cell end table close square brackets space space open square brackets table row 1 cell space space space space 0 end cell cell space space space 2 end cell row 0 cell space space space space 2 end cell cell space space 1 end cell row 2 cell space space space 0 end cell cell space space 3 end cell end table close square brackets space space space space space equals space space open square brackets table row cell 5 plus 0 plus 16 end cell cell space space space space 0 plus 0 plus 0 end cell cell space space space 10 plus 0 plus 24 end cell row cell 2 plus 0 plus 16 end cell cell space space space space 0 plus 8 plus 0 end cell cell space space 4 plus 4 plus 15 end cell row cell 8 plus 0 plus 26 end cell cell space space space space 0 plus 0 plus 0 end cell cell space space 16 plus 0 plus 39 end cell end table close square brackets space equals space open square brackets table row cell 21 space space end cell 0 cell space space 34 end cell row cell 12 space space end cell 8 cell space space 23 end cell row 34 0 cell space space 55 end cell end table close square brackets C o n s i d e r space straight A cubed space minus 6 straight A plus 7 straight A plus 21 space space space space space equals space open square brackets table row cell 21 space space end cell 0 cell space space 34 end cell row cell 12 space space end cell 8 cell space space 23 end cell row 34 0 cell space space 55 end cell end table close square brackets minus 6 space space open square brackets table row 5 cell space space space space 0 end cell cell space space space 8 end cell row 2 cell space space space space 4 end cell cell space space space 5 end cell row 8 cell space space space space 0 end cell cell space space space 13 end cell end table close square brackets plus 7 space open square brackets table row 1 cell space space space space space 0 end cell cell space space space 2 end cell row 0 cell space space space space space 2 end cell cell space space 1 end cell row 2 cell space space space space 0 end cell cell space space 3 end cell end table close square brackets plus 2 space open square brackets table row 1 cell space space space space 0 end cell cell space space space 0 end cell row 0 cell space space space 1 end cell cell space space space 0 end cell row 0 cell space space space 0 end cell cell space space space 1 end cell end table close square brackets space space space space equals space space open square brackets table row cell 21 space space end cell 0 cell space space 34 end cell row cell 12 space space end cell 8 cell space space 23 end cell row 34 0 cell space space 55 end cell end table close square brackets space plus space open square brackets table row cell negative 30 end cell cell space space space 0 end cell cell space space minus 48 end cell row cell negative 12 space space space end cell cell negative 24 end cell cell space space minus 30 end cell row cell negative 48 end cell cell space space 0 end cell cell space minus 78 end cell end table close square brackets plus space open square brackets table row 7 cell space space space 0 end cell cell space space space 14 end cell row 0 cell space space space 14 end cell cell space space space 7 end cell row 14 cell space space space 0 end cell cell space space space 21 end cell end table close square brackets plus space open square brackets table row 2 cell space space space 0 end cell cell space space space 0 end cell row 0 cell space space space 2 end cell cell space space 0 end cell row cell 0 space end cell cell space space space 0 end cell cell space space 2 end cell end table close square brackets space space space equals space space open square brackets table row cell 21 minus 30 plus 7 plus 2 end cell cell space space space space space space space space space space space space space space space space 0 plus 0 plus 0 plus 0 end cell cell space space space space space space space space space 34 minus 48 plus 14 plus 0 end cell row cell 12 minus 12 plus 0 plus 0 end cell cell space space space space space space space space space space 8 minus 24 plus 14 plus 2 end cell cell space space space space space space space space space space space 23 minus 30 plus 7 plus 0 end cell row cell 34 minus 48 plus 14 plus 0 end cell cell space space space space space space space space space space space space space space space space space space space space space 0 plus 0 plus 0 plus 0 end cell cell space space space space space space space space space space space 55 minus 78 plus 21 plus 2 end cell end table close square brackets space space space space equals space space open square brackets table row 0 cell space space space 0 end cell cell space space space 0 end cell row 0 cell space space space 0 end cell cell space space space space 0 end cell row 0 cell space space space 0 end cell cell space space space 0 end cell end table close square brackets equals space straight O therefore space space straight A cubed space minus 6 straight A squared plus 7 straight A plus 2 space straight I space equals space straight O

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