M is a matrix with real entries given by `M=[{:(4,k,0),(6,3,0),(2,1,k):}]` Which of the following conditons guarantee the invertibility of M ?
1. `k ne 2`
2. `k ne 0`
3. `k ne 1`
4. `k = 1`
select the correct answer using the code given below :

Options

A. A. 1 and 2
B. B. 2 and 3
C. C. 1 and 4
D. D. 3 and 4

Answer

To determine the conditions that guarantee the invertibility of the matrix \( M \), we need to find its determinant and ensure that it is not equal to zero.

Given the matrix:
\[ M = \begin{pmatrix} 4 & k & 0 \\ 6 & 3 & 0 \\ 2 & 1 & k \end{pmatrix} \]

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

To find the determinant of a 3x3 matrix, we can use the formula:
\[ \text{det}(M) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix \( M \):
- \( a = 4, b = k, c = 0 \)
- \( d = 6, e = 3, f = 0 \)
- \( g = 2, h = 1, i = k \)

Using the determinant formula, we can simplify:
\[ \text{det}(M) = 4(3k - 0) - k(6k - 0) + 0 \]
\[ = 12k - 6k^2 \]

### Step 2: Set the Determinant Not Equal to Zero

For the matrix to be invertible, we need:
\[ 12k - 6k^2 \neq 0 \]

### Step 3: Factor the Expression

We can factor out common terms:
\[ 6k(2 - k) \neq 0 \]

### Step 4: Determine Conditions from the Factored Expression

From the factored expression \( 6k(2 - k) \neq 0 \), we can derive the conditions:
1. \( 6k \neq 0 \) implies \( k \neq 0 \)
2. \( 2 - k \neq 0 \) implies \( k \neq 2 \)

Thus, the conditions that guarantee the invertibility of \( M \) are:
- \( k \neq 0 \)
- \( k \neq 2 \)

### Conclusion

The correct options that guarantee the invertibility of \( M \) are:
1. \( k \neq 2 \)
2. \( k \neq 0 \)

### Selected Answer

The correct answer is options 1 and 2.

---