For what value of `k`, inverse does not exist for the matrix `[(1,2),(k,6)]`

Options

A. a) 0
B. b) 3
C. c) 6
D. d) 2

Answer

To determine the value of \( k \) for which the inverse of the matrix

\[ \begin{pmatrix} 1 & 2 \\ k & 6 \end{pmatrix} \]

does not exist, we need to find when the determinant of the matrix is equal to zero.

### Step-by-step Solution:

1. **Write down the matrix**:
The matrix is given as

\[ A = \begin{pmatrix} 1 & 2 \\ k & 6 \end{pmatrix} \]

2. **Calculate the determinant**:
The determinant of a \( 2 \times 2 \) matrix

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

is calculated using the formula \( \text{det}(A) = ad - bc \).

For our matrix, we have:
- \( a = 1 \)
- \( b = 2 \)
- \( c = k \)
- \( d = 6 \)

Therefore, the determinant is:

\[ \text{det}(A) = (1)(6) - (2)(k) = 6 - 2k \]

3. **Set the determinant to zero**:
To find when the inverse does not exist, we set the determinant equal to zero:

\[ 6 - 2k = 0 \]

4. **Solve for \( k \)**:
Rearranging the equation gives:

\[ 2k = 6 \]

Dividing both sides by 2:

\[ k = 3 \]

5. **Conclusion**:
The value of \( k \) for which the inverse of the matrix does not exist is

\[ \boxed{3} \]