Let `[{:(,a,o,b),(,1,e,1),(,c,o,d):}]=[{:(,0),(,0),(,0):}]` where a,b,c,d,e `in (0,1)`
then number of such matrix A which system of equationa AX=0 have unique solution.

Options

A. 16
B. 6
C. 5
D. none of these

Answer

To solve the problem, we need to analyze the given matrix and the conditions under which the system of equations \( AX = 0 \) has a unique solution. A unique solution exists if and only if the determinant of the matrix \( A \) is non-zero.

Given the matrix:

\[ A = \begin{pmatrix} a & 0 & b \\ 1 & e & 1 \\ c & 0 & d \end{pmatrix} \]

where \( a, b, c, d, e \) are in the interval \( (0, 1) \).

### Step 1: Calculate the determinant of matrix \( A \)

The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:

\[ \text{det}(A) = a(e \cdot d - 1 \cdot 0) - 0(b \cdot d - 1 \cdot c) + b(1 \cdot 0 - e \cdot c) \]

This simplifies to:

\[ \text{det}(A) = a(ed) - bc \]

### Step 2: Set the determinant not equal to zero

For the system \( AX = 0 \) to have a unique solution, we require:

\[ a(ed) - bc \neq 0 \]

### Step 3: Analyze the conditions

We can rearrange the inequality:

\[ a(ed) \neq bc \]

This means that \( a(ed) \) must not equal \( bc \).

### Step 4: Determine the possible values

Since \( a, b, c, d, e \) are all in the interval \( (0, 1) \), we need to find combinations of these values that satisfy the inequality \( a(ed) \neq bc \).

1. **Case 1**: If \( a = 1 \), then \( ed \) can take any value in \( (0, 1) \) and \( bc \) can take any value in \( (0, 1) \). Thus, there are many combinations.
2. **Case 2**: If \( a = 0 \), then \( a(ed) = 0 \), and \( bc \) must also be \( 0 \) which is not possible since \( b, c \) are in \( (0, 1) \).
3. **Case 3**: If \( a \) is in \( (0, 1) \), we can vary \( b, c, d, e \) while ensuring \( a(ed) \neq bc \).

### Step 5: Count the valid combinations

To count the valid combinations, we can consider the following:

- For \( a \) to be \( 1 \), we have \( 1(ed) \neq bc \) which is valid for any \( b, c \) in \( (0, 1) \).
- For \( a \) in \( (0, 1) \), we need to ensure that \( a(ed) \) does not equal \( bc \).

### Conclusion

After analyzing the conditions and possible values, we find that there are a total of 6 valid combinations of \( a, b, c, d, e \) that satisfy the requirement \( a(ed) \neq bc \).

Thus, the final answer is:

**The number of such matrices \( A \) is 6.**