If $A=\left[\begin{array}{ccc}1& 1& -2\\ 0& 1& 4\\ 2& 3& 1\end{array}\right] and A^3-3A^2-5A=kI then k = .............$

$A^2=\left[\begin{array}{ccc}1& 1& -2\\ 0& 1& 4\\ 2& 3& 1\end{array}\right] \left[\begin{array}{ccc}1& 1& -2\\ 0& 1& 4\\ 2& 3& 1\end{array}\right]=\left[\begin{array}{ccc}1+0-4& 1+1-6& -2+4-2\\ 0+0+8& 0+1+12& 0+4+4\\ 2+0+2& 2+3+3& -4+12+1\end{array}\right]=\left[\begin{array}{ccc}-3& -4& 0\\ 8& 13& 8\\ 4& 8& 9\end{array}\right]$

$A^3=A A^2=\left[\begin{array}{ccc}1& 1& -2\\ 0& 1& 4\\ 2& 3& 1\end{array}\right] \left[\begin{array}{ccc}1& 1& -2\\ 8& 13& 8\\ 4& 8& 9\end{array}\right]=\left[\begin{array}{ccc}-3+8-8& -4+13-16& 0+8-18\\ 0+8+16& 0+13+32& 0+8+36\\ -6+24+4& -8+39+8& 0+24+9\end{array}\right]=\left[\begin{array}{ccc}-3& -7& -10\\ 24& 45& 44\\ 22& 39& 33\end{array}\right]$

$A^3-3A^2-5A-I=\left[\begin{array}{ccc}-3& -7& -10\\ 24& 45& 44\\ 22& 39& 33\end{array}\right]-3 \left[\begin{array}{ccc}-3& -4& 0\\ 8& 13& 8\\ 4& 8& 9\end{array}\right]=5 \left[\begin{array}{ccc}1& 1& -2\\ 0& 1& 4\\ 2& 3& 1\end{array}\right]-\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

= $\left[\begin{array}{ccc}-3+9-5-1& -7+12-5-0& -10-0+10-0\\ 24-24-0-0& 45-39-5-1& 44-24-20-0\\ 22-12-10-0& 39-24-15-0& 33-27-5-1\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]=O$

$\Rightarrow KI-I=O\Rightarrow k=1.$