Total Number of 2×2 Matrices with Entries 0 and 1

Step 1: Understand the Matrix

\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

Step 2: Possible Values

Each entry can be either 0 or 1.

\[ \text{Choices for each entry} = 2 \]

Step 3: Total Entries

\[ \text{Total entries} = 4 \]

Step 4: Total Matrices

\[ \text{Total} = 2^4 = 16 \]

Final Answer:

\[ \boxed{16} \]