Question

If $f\left(x\right)=x^2-5x+6$, find $f\left(A\right)$ if $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$.

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

Answer 1

The correct option is A $\left[\begin{array}{ccc}1& -1& -3\\ 1& 1& 10\\ -5& 4& -4\end{array}\right]$

$f\left(x\right)=x^2-5x+6$

$f\left(A\right)=A^2-5A+6$

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

$\begin{array}{cc}{A}^2& =\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]\\ & =\left[\begin{array}{ccc}4+0+1& 0+0-1& 2+0+0\\ 4+2+3& 0+1-3& 2+3+0\\ 2-2+0& 0-1+0& 1-3+0\end{array}\right]\end{array}$

$=\left[\begin{array}{ccc}5& -1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]$

$5A=\left[\begin{array}{ccc}10& 0& 5\\ 10& 5& 15\\ 5& -5& 0\end{array}\right]$

On Putting in $f\left(A\right)$

$\begin{array}{cc}f\left(A\right)& =\left[\begin{array}{ccc}5& -1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]-\left[\begin{array}{ccc}10& 0& 5\\ 10& 5& 15\\ 5& -5& 0\end{array}\right]+\left[\begin{array}{ccc}6& 0& 0\\ 0& 6& 0\\ 0& 0& 6\end{array}\right]\\ & =\left[\begin{array}{ccc}5-10+6& -1+0+0& 2-5+0\\ 9-10+0& -2-5+6& 5-15+0\\ 0-5+0& -1+5+0& -2+0+6\end{array}\right]\\ & =\left[\begin{array}{ccc}1& -1& -3\\ -1& -1& -10\\ -5& 4& -4\end{array}\right]\end{array}$