Question

If $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]$ is a root of polynomial $x^3-6x^2+7x+k=0$, then the value of $k$ is :

• A. $2$
• B. $4$
• C. $-2$
• D. $1$

Answer 1

The correct option is A $2$

$A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]$ is a root of polynomial $x^3-6x^2+7x+k$

Since, we know the matrix, we can find the characteristics polynomial of this matrix, $p\left(\lambda \right)$ when $\lambda$ are the eigenvalue of the matrix which are the roots of polynomial.

$\Rightarrow p\left(\lambda \right)=det|A-\lambda I|,$ where $I$ is identity matrix.

$\Rightarrow \left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]-\lambda \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1-\lambda & 0& 2\\ 0& 2-\lambda & 1\\ 2& 0& 3-\lambda \end{array}\right]$

Determinant : $|(1-\lambda )[(2-\lambda )(3-\lambda )-0]+0[2-0]+2[0-2(2-\lambda )]|$

$=|(1-\lambda )[6-5\lambda +{\lambda }^2]+0+2(-4+2\lambda )|$

$=|6-5\lambda +{\lambda }^2-6\lambda +5{\lambda }^2-{\lambda }^3-8+4\lambda |$

$=|-{\lambda }^3+6{\lambda }^2-7\lambda -2|$

We see, that the characteristic polynomial is same as the polynomial given. If we compare both, $k=2.$

Hence, the answer is $2.$