$$A = \begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} \Rightarrow A^{2019} =$$

$$A = \begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix}$$

$$A^{2} = \begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} * \begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} = \begin{bmatrix}2 & 2 \\2 & 2 \end{bmatrix}$$

$$A^{2} = 2\begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} = 2A$$

$$A = 2$$

$$A^{4} = \begin{bmatrix}2 & 2 \\2 & 2 \end{bmatrix} * \begin{bmatrix}2 & 2 \\2 & 2 \end{bmatrix} = \begin{bmatrix}8 & 8 \\8 & 8 \end{bmatrix}$$

$$A^{4} = (2A)^{2} = 8\begin{bmatrix}1 & 1 \\1 & 1 \end{bmatrix} = 8A$$[$$Since, A^{2} = 2A$$]

$$A^{4} = 4A^{2} = 8A = 2^{2}A^{2}$$

$$Now, (A^{4})^{504} = (2^{2})^{504}(A^{2})^{504}$$

$$A^{2016} = 2^{1008}(2^{2})^{504}$$[$$Since; A = 2$$]

$$A^{2016} = 2^{2016}$$

$$A^{2016}A^{3} = 2^{2016}A^{3}$$

$$A^{2019} = 2^{2016}A^{2}A$$

$$A^{2019} = 2^{2016}2^{2}A$$[$$Since; A = 2$$]

$$A^{2019} = 2^{2018}A$$