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

Options

A. 0
B. 3
C. 6
D. 2

Answer

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

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

does not exist, we need to find when the determinant of the matrix \( A \) is equal to zero. The determinant of a \( 2 \times 2 \) matrix

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

is calculated using the formula:

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

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

For our matrix \( A \):

- \( a = 1 \)
- \( b = 2 \)
- \( c = k \)
- \( d = 6 \)

Using the determinant formula:

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

### Step 2: Set the determinant equal to zero

To find the value of \( k \) for which the inverse does not exist, we set the determinant equal to zero:

\[ 6 - 2k = 0 \]

### Step 3: Solve for \( k \)

Now, we solve the equation:

\[ 6 - 2k = 0 \]

Rearranging gives:

\[ 2k = 6 \]

Dividing both sides by 2:

\[ k = \frac{6}{2} = 3 \]

### Conclusion

Thus, the value of \( k \) for which the inverse of the matrix does not exist is:

\[ \boxed{3} \]