Question

If $A=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& -1\\ 3& -1& 1\end{array}\right]$, then

• A. $A^3+3A^2+A-9I_3=0$
• B. $A^3-3A^2+A+9I_3=0$
• C. $A^3+3A^2-A+9I_3=0$
• D. $A^3-3A^2-A+9I_3=0$

Answer 1

The correct option is D

$A^3-3A^2-A+9I_3=0$

$\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& -1\\ 3& -1& 1\end{array}\right]A^2=A.A=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& -1\\ 3& -1& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& -1\\ 3& -1& 1\end{array}\right]=\left[\begin{array}{ccc}4& 3& 0\\ -3& 2& -2\\ 6& 4& 5\end{array}\right]A.A^2=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& -1\\ 3& -1& 1\end{array}\right]\left[\begin{array}{ccc}4& 3& 0\\ -9& -2& -7\\ 21& 11& 7\end{array}\right]=\left[\begin{array}{ccc}4& 11& 1\\ -9& -2& -7\\ 21& 11& 7\end{array}\right]\Rightarrow A^3-3A^2-A+9I_3=0$