Question

If $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right],$ then $A^2-5A+6I=$

• A. $\left[\begin{array}{ccc}1& -1& -5\\ -1& -1& 4\\ -3& -10& 4\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& -1& -3\\ -1& -1& -10\\ -5& 4& 4\end{array}\right]$
• C. $0$
• D. $I$

Answer 1

Answer: (B)

$A^2-5A+6I=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$
$-\left[\begin{array}{ccc}10& 0& 5\\ 10& 5& 15\\ 5& -5& 0\end{array}\right]+\left[\begin{array}{ccc}6& 0& 0\\ 0& 6& 0\\ 0& 0& 6\end{array}\right]$
$=\left[\begin{array}{ccc}5& -1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]-\left[\begin{array}{ccc}10& 0& 5\\ 10& 5& 15\\ 5& -5& 0\end{array}\right]+\left[\begin{array}{ccc}6& 0& 0\\ 0& 6& 0\\ 0& 0& 6\end{array}\right]$
$=\left[\begin{array}{ccc}-5& -1& -3\\ -1& -7& -10\\ -5& 4& -2\end{array}\right]+\left[\begin{array}{ccc}6& 0& 0\\ 0& 6& 0\\ 0& 0& 6\end{array}\right]=\left[\begin{array}{ccc}1& -1& -3\\ -1& -1& -10\\ -5& 4& 4\end{array}\right]$