Question

If ${\left[\begin{array}{ccc}1& 2& a\\ 0& 1& 4\\ 0& 0& 1\end{array}\right]}^n=\left[\begin{array}{ccc}1& 18& 2007\\ 0& 1& 36\\ 0& 0& 1\end{array}\right]$ $($where $[.]$ denotes the greatest integer function and $n\in N),$ then the value of $\left[\frac{n+a}{100}\right]$ is

Answer 1

Let $A=\left[\begin{array}{ccc}1& 2& a\\ 0& 1& 4\\ 0& 0& 1\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{ccc}1& 2& a\\ 0& 1& 4\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& a\\ 0& 1& 4\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 4& 2a+8\\ 0& 1& 8\\ 0& 0& 1\end{array}\right]$
$A^3=\left[\begin{array}{ccc}1& 6& 3a+24\\ 0& 1& 12\\ 0& 0& 1\end{array}\right]$
$A^4=\left[\begin{array}{ccc}1& 8& 4a+48\\ 0& 1& 16\\ 0& 0& 1\end{array}\right]$
$\cdot \cdot$
$\cdot \cdot$
$\cdot \cdot$
$A^n=\left[\begin{array}{ccc}1& 2n& na+8\sum _{k=0}^{n-1}k\\ 0& 1& 4n\\ 0& 0& 1\end{array}\right]$

Now,${\left[\begin{array}{ccc}1& 2& a\\ 0& 1& 4\\ 0& 0& 1\end{array}\right]}^n=\left[\begin{array}{ccc}1& 18& 2007\\ 0& 1& 36\\ 0& 0& 1\end{array}\right]\Rightarrow \left[\begin{array}{ccc}1& 2n& na+8\sum _{k=0}^{n-1}k\\ 0& 1& 4n\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 18& 2007\\ 0& 1& 36\\ 0& 0& 1\end{array}\right]$

$\Rightarrow 2n=18\Rightarrow n=9\Rightarrow na+8\sum _{k=0}^{n-1}k=2007\Rightarrow 9a+8\sum _{k=0}^{8}k=2007\Rightarrow 9a+8\times 36=2007\Rightarrow 9a=1719\Rightarrow a=191$

$\therefore \left[\frac{9+191}{100}\right]=\left[2\right]=2$