Question

If $A=\left[\begin{array}{cc}0& 5\\ 0& 0\end{array}\right]$ and $f\left(x\right)=1+x+x^2+......+x^6$, then $f\left(A\right)=$

• A. $0$
• B. $\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 5\\ 0& 0\end{array}\right]$
• D. $\left[\begin{array}{cc}0& 5\\ 1& 1\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]$

$A=\left[\begin{array}{cc}0& 5\\ 0& 0\end{array}\right]A^2=\left[\begin{array}{cc}0& 5\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 5\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]A^3=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 5\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
Similarly $A^4=A^5=A^6=0$
Now
As $f\left(x\right)=1+x+x^2+......+x^6$
$f\left(A\right)=1+\left[\begin{array}{cc}0& 5\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\left[\begin{array}{cc}0& 5\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]$