The number of 3 x 3 non-singular matrices, with four entries as 1 and all other entries as 0, is:- (1) 5 (2) 6 (3) at least 7 (4) less than 4

Options

A. 5
B. 6
C. at least 7
D. less than 4

Explanation

To find the number of 3 x 3 non-singular matrices with four entries as 1 and all other entries as 0, we can follow these steps:

### Step 1: Understand the Matrix Structure
A 3 x 3 matrix has 9 entries. If we want four entries to be 1, that means there will be 5 entries that are 0. The matrix must be non-singular, meaning it must have a non-zero determinant.

### Step 2: Identify the Non-Singular Condition
For a 3 x 3 matrix to be non-singular, the rows (or columns) must be linearly independent. With four 1s and five 0s, we need to ensure that no row or column is entirely filled with zeros.

### Step 3: Consider the Placement of 1s
We can place the four 1s in such a way that no row or column is completely zero. We can start by placing three 1s in different rows and columns, which guarantees that the matrix remains non-singular.

### Step 4: Count the Arrangements
1. **Choose 3 positions for 1s**: We can select 3 positions for the 1s from the 9 available positions. This can be done in \( \binom{9}{3} \) ways.
2. **Place the 4th 1**: After placing the first three 1s, we have to place the fourth 1. The fourth 1 can be placed in any of the remaining positions that do not create a row or column of all zeros.

### Step 5: Calculate the Combinations
- The number of ways to choose 3 positions from 9 is given by:
\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

- For each of these arrangements, we need to ensure that the fourth 1 is placed in a position that maintains the non-singularity of the matrix.

### Step 6: Final Count
After considering the placements and ensuring that the matrix remains non-singular, we find that there are a total of 6 valid configurations for the placement of the fourth 1.

### Conclusion
Thus, the total number of 3 x 3 non-singular matrices with four entries as 1 and all other entries as 0 is **6**.