If A, = ,left[ begin{gathered}
1 ,1 | vedclass

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Question

$|A-\lambda I|=0$

$\left| {\begin{array}{*{20}{c}}
{1 - \lambda }&1&2\
0&{2 - \lambda }&1\
1&0&{2 - \lambda }
\end{ar

Vedclass · Accepted Answer

$6$

Vedclass · Answer

$|A-\lambda I|=0$

$\left| {\begin{array}{*{20}{c}}
{1 - \lambda }&1&2\
0&{2 - \lambda }&1\
1&0&{2 - \lambda }
\end{array}} ight| = 0$

$(1-\lambda)\left[(2-\lambda)^{2}-0ight]-(0-1)+2(0-(2-\lambda))=0$

$(1-\lambda)(2-\lambda)^{2}+1-4+2 \lambda=0$

$(1-\lambda)\left(\lambda^{2}-4 \lambda+4ight)-3+2 \lambda=0$

$\lambda^{2}-4 \lambda+4-\lambda^{3}+4 \lambda^{2}-4 \lambda-3+2 \lambda=0$

$\lambda^{3}=5 \lambda^{2}-6 \lambda+1=(5 \lambda-1)(\lambda-1)$

$ A^{3}=(5 A-I)(A-I) $

$ a=5, b=1 $ or $ a=1, b=5 $

$ \Rightarrow a+b=6 $

If $A\, = \,\left[ \begin{gathered}
1\ \ \ \,1\ \ \ \,2\ \ \ \hfill \\
0\ \ \ \,2\ \ \ \,1\ \ \ \hfill \\
1\ \ \ \,0\ \ \ \,2\ \ \ \hfill \\
\end{gathered} \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to

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