$A = \left[ {\begin{array}{*{20}{c}}1&0\\1&1\end{array}} \right]$ and $I = \left[ {\begin{array}{*{20}{c}}
1&0\\0&1\end{array}} \right]$ , then which of the following holds for all $n \geq 2, n \in N$ ?

If $A = \left( {\begin{array}{*{20}{c}}2&{ – 1}\\{ – 1}&2\end{array}} \right)$ and $I$ is the unit matrix of order $2$, then ${A^2}$ equals

If the matrix $\left[ {\begin{array}{*{20}{c}}1&3&{\lambda + 2}\\2&4&8\\3&5&{10}\end{array}} \right]$ is singular, then $\lambda = $

Let $A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$. If $B=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A \left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$ then the sum of all the elements of the matrix $\sum \limits_{n=1}^{50} B^n$ is equal to

If $A = \left[ {\begin{array}{*{20}{c}}
1&0\\
{\frac{1}{2}}&1
\end{array}} \right]$ , then $A^{50}$ is