$A-2 I=\left[\begin{array}{ccc}1 & 0 & 4 \\ 0 & 1 & -2 \\ 1 & 0 & 2\end{array}\right]$

Step 1 ：The problem is asking to find the matrix A given that A - 2I equals the given matrix. Here, I is the identity matrix. The identity matrix is a square matrix in which all the elements of the principal (main) diagonal are ones and all other elements are zeros. For a 3x3 matrix, the identity matrix is: \[ I = \left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] \]

Step 2 ：So, 2I would be: \[ 2I = \left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right] \]

Step 3 ：To find A, we need to add 2I to the given matrix. This is because A - 2I = given matrix implies that A = 2I + given matrix.

Step 4 ：Given matrix is: \[ \text{given_matrix} = \left[\begin{array}{ccc} 1 & 0 & 4 \\ 0 & 1 & -2 \\ 1 & 0 & 2\end{array}\right] \]

Step 5 ：Adding 2I to the given matrix, we get: \[ A = \left[\begin{array}{ccc} 3 & 0 & 4 \\ 0 & 3 & -2 \\ 1 & 0 & 4\end{array}\right] \]

Step 6 ：Final Answer: The matrix A is \[ A = \boxed{\left[\begin{array}{ccc}3 & 0 & 4 \\ 0 & 3 & -2 \\ 1 & 0 & 4\end{array}\right]} \]