<s>[INST] Find Matrix $A^{50}$? Find the matrix $A^{50}$ given
$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$
I was practicing some questions for my exam and I found questions of this form in a previous year's paper.
I don't know how to do such questions.
Please assist over this question.
Thank You
 [/INST] You can also compute it with the following method. First note that $50 = 32 + 16 + 2$. Let $$A=\left(\begin{array}{cc}2&0\\2&1\end{array}\right).$$
Then $$A^{50} = A^{32} A^{16} A^2$$.
We can compute $A^{2^n}$ easier than $A^{50}$,
$$A^2 = \left( \begin{array}{cc} 4 & 0\\ 6 & 1 \end{array} \right)$$
$$A^4 = \left( \begin{array}{cc} 4 & 0\\ 6 & 1 \end{array} \right)\left( \begin{array}{cc} 4 & 0\\ 6 & 1 \end{array} \right)= \left( \begin{array}{cc} 16 & 0\\ 30 & 1 \end{array} \right)$$
$$A^8 =  \left( \begin{array}{cc} 16 & 0\\ 30 & 1 \end{array} \right) \left( \begin{array}{cc} 16 & 0\\ 30 & 1 \end{array} \right) =  \left( \begin{array}{cc} 256 & 0\\ 510 & 1 \end{array} \right)$$
$$A^{16} = \left( \begin{array}{cc} 256 & 0\\ 510 & 1 \end{array} \right)\left( \begin{array}{cc} 256 & 0\\ 510 & 1 \end{array} \right)=\left( \begin{array}{cc} 65536& 0\\ 131070& 1 \end{array} \right)$$
and
$$A^{32} = \left( \begin{array}{cc} 65536& 0\\ 131070& 1 \end{array} \right)\left( \begin{array}{cc} 65536& 0\\ 131070& 1 \end{array} \right)=\left( \begin{array}{cc} 4294967296& 0\\ 8589934590 & 1 \end{array} \right).$$
Finally $$A^{50} =\left( \begin{array}{cc} 4294967296& 0\\ 8589934590 & 1 \end{array} \right) \left( \begin{array}{cc} 65536& 0\\ 131070& 1 \end{array} \right) \left( \begin{array}{cc} 4 & 0\\ 6 & 1 \end{array} \right).$$
This isn't the best way to go about it, but it is a lot easier than multiplying $A$ to itself 50 times. You should probably persue @science's method to be honest.
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