If A = [ * 20 c 1&2&1 0&1& - 1 3& - 1 &1 ], then

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MCQ

If $A = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]$, then

A
${A^3} + 3{A^2} + A - 9{I_3} = O$
B
${A^3} - 3{A^2} + A + 9{I_3} = O$
C
${A^3} + 3{A^2} - A + 9{I_3} = O$
✓
${A^3} - 3{A^2} - A + 9{I_3} = O$

✓

Answer

Correct option: D.

${A^3} - 3{A^2} - A + 9{I_3} = O$

d
(d) $A = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]$

${A^2} = A\,.\,A = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]\,$$ = \left[ {\begin{array}{*{20}{c}}4&3&0\\{ - 3}&2&{ - 2}\\6&4&5\end{array}} \right]$

$A\,.\,{A^2} = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}4&3&0\\{ - 3}&2&{ - 2}\\6&4&5\end{array}} \right]\, = \left[ {\begin{array}{*{20}{c}}4&{11}&1\\{ - 9}&{ - 2}&{ - 7}\\{21}&{11}&7\end{array}} \right]$

==> ${A^3} - 3{A^2} - A + 9{I_3} = 0$.

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