Question

If $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\\ 2& 1& 0\end{array}\right],I$ is the unit matrix of order 3 and $a{A}^3+b{A}^2+cA–I=B$, where B is null matrix of order $3\times 3$, then

• A. $a+b+c+3=0$
• B. $2a+2b+c+3=0$
• C. $2a+2b=0$
• D. $3a+c=0$

Answer 1

Answer: (A), (B), (C), (D)

$3a+c=0$

$|A–\lambda I|=0$
$\Rightarrow \left|\begin{array}{ccc}1\lambda & 1& 1\\ 1& \lambda & 2\\ 2& 1& \lambda \end{array}\right|=0$
$\Rightarrow (1–\lambda )[{\lambda }^2–2]–1(\lambda –4)+1(–1+2\lambda )=0$
$\Rightarrow {\lambda }^3–\lambda 2–3\lambda –1=0$
$\Rightarrow A^3–A^2–3A–I=0$
$\Rightarrow a=1,b=–1,c=–3$
Hence all option are true.