Question

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

• A. 4
• B. 5
• C. 6
• D. 7

Answer 1

The correct option is C

6

$|A-\lambda I|=0\left|\begin{array}{ccc}1-\lambda & 1& 2\\ 0& 2-\lambda & 1\\ 1& 0& 2-\lambda \end{array}\right|=0\Rightarrow (1-\lambda )\left[\right(2-\lambda {)}^2-0]-(0-1)+2(0-(2-\lambda ))=0\Rightarrow (1-\lambda \left)\right(2-\lambda {)}^2+1-4+2\lambda =0\Rightarrow (1-\lambda )({\lambda }^2-4\lambda +4)-3+2\lambda =0\Rightarrow {\lambda }^3=5{\lambda }^2-6\lambda +1=(5\lambda -1)(\lambda -1)A^3=(5A-I)(A-I)\{\therefore \lambda =A\}a=5,b=1ora=1,b=5$