Question

If $A=\left[\begin{array}{ccc}6& 2& 2\\ -2& -3& -1\\ 2& -1& 3\end{array}\right]$, then $|xI-A|$ equals

• A. $x^3+12x^2-36x+12$
• B. $x^3-12x^2+36x-12$
• C. $x^3-12x^2-36x+12$
• D. none of these

Answer 1

Answer: (D)

none of these

$A=\left[\begin{array}{ccc}6& 2& 2\\ -2& -3& -1\\ 2& -1& 3\end{array}\right]$

$|xI-A|=\left[\begin{array}{ccc}x-6& -2& -2\\ 2& x+3& 1\\ -2& 1& x-3\end{array}\right]$

applying $R_3\to R_2+R_3$ and expanding gives

$|xI-A|=\left[\begin{array}{ccc}x-6& -2& -2\\ 2& x+3& 1\\ 0& x+4& x-2\end{array}\right]$

$=x^3-6x^2-10x+36$

Hence, option D is correct.