$$A = \begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix}$$
$$A^{2} = \begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} *Â \begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} =Â \begin{bmatrix}2 & 2 \\2 & 2 \end{bmatrix}$$
$$A^{2} = 2\begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} = 2A$$
$$A = 2$$
$$A^{4} = \begin{bmatrix}2 & 2 \\2 & 2 \end{bmatrix} * \begin{bmatrix}2 & 2 \\2 & 2 \end{bmatrix} = \begin{bmatrix}8 & 8 \\8 & 8 \end{bmatrix}$$
$$A^{4} = (2A)^{2} =Â 8\begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} = 8A$$[$$Since, A^{2} = 2A$$]
$$A^{4} = 4A^{2} = 8A = 2^{2}A^{2}$$
$$Now, (A^{4})^{504} =Â (2^{2})^{504}(A^{2})^{504}$$
$$A^{2016} = 2^{1008}(2^{2})^{504}$$[$$Since; A = 2$$]
$$A^{2016} = 2^{2016}$$
$$A^{2016}A^{3} = 2^{2016}A^{3}$$
$$A^{2019} = 2^{2016}A^{2}A$$
$$A^{2019} = 2^{2016}2^{2}A$$[$$Since; A = 2$$]
$$A^{2019} = 2^{2018}A$$
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