Consider a matrix `M=[{:(3,4,0),(2,1,0),(3,1,k):}]` and the following statements
Statement A : Inverse of M exists.
Statement B : k `ne` 0
Which one of the following in respect of the above matrix and statement is correct ?

Options

A. A implies B, but B does not imply A
B. B implies A, but does not imply B
C. Neither A implies B nor B implies A
D. A implies B as well as B implies A

Explanation

To determine the correctness of the statements regarding the matrix \( M = \begin{pmatrix} 3 & 4 & 0 \\ 2 & 1 & 0 \\ 3 & 1 & k \end{pmatrix} \), we need to analyze the determinant of the matrix and the implications of the statements.

### Step 1: Calculate the Determinant of Matrix \( M \)

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

\[ \text{det}(M) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( M \):

\[ M = \begin{pmatrix} 3 & 4 & 0 \\ 2 & 1 & 0 \\ 3 & 1 & k \end{pmatrix} \]

Using the determinant formula, we have:

\[ \text{det}(M) = 3 \cdot (1 \cdot k - 0 \cdot 1) - 4 \cdot (2 \cdot k - 0 \cdot 3) + 0 \cdot (2 \cdot 1 - 1 \cdot 3) \]

This simplifies to:

\[ \text{det}(M) = 3k - 4(2k) = 3k - 8k = -5k \]

### Step 2: Analyze the Determinant

For the inverse of the matrix \( M \) to exist, the determinant must not be equal to zero:

\[ -5k \neq 0 \]

This implies:

\[ k \neq 0 \]

### Step 3: Evaluate the Statements

- **Statement A**: The inverse of \( M \) exists. This is true if \( k \neq 0 \).
- **Statement B**: \( k \neq 0 \). This is also true if the inverse exists.

### Conclusion

- If \( k \neq 0 \) (Statement B), then \( \text{det}(M) \neq 0 \), which means the inverse exists (Statement A).
- Conversely, if the inverse exists (Statement A), it implies \( k \neq 0 \) (Statement B).

Thus, both statements imply each other, leading us to conclude that:

**Both Statement A and Statement B are correct, and they imply each other.**

### Final Answer

The correct option is that both statements imply each other.

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