The number of 2 `x` 2 matrices A such that all entries are either 1 or 0 is

Options

A. 12
B. 6
C. 9
D. 16

Answer

To find the number of 2x2 matrices \( A \) such that all entries are either 1 or 0, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure of the Matrix**:
A 2x2 matrix has 4 entries. We can represent a 2x2 matrix as follows:
\[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \]
where each entry \( a_{ij} \) can either be 0 or 1.

2. **Determine the Number of Choices for Each Entry**:
For each of the 4 entries in the matrix, we have 2 choices (either 0 or 1).

3. **Calculate the Total Number of Matrices**:
Since each of the 4 entries can be independently chosen to be either 0 or 1, the total number of different matrices can be calculated using the formula:
\[ \text{Total Matrices} = 2^{\text{number of entries}} = 2^4 \]

4. **Perform the Calculation**:
Now, we compute \( 2^4 \):
\[ 2^4 = 16 \]

5. **Conclusion**:
Therefore, the total number of 2x2 matrices \( A \) such that all entries are either 1 or 0 is \( 16 \).

### Final Answer:
The number of 2x2 matrices \( A \) such that all entries are either 1 or 0 is \( 16 \).