If matrix `A=[{:(3,2,4),(1,2,-1),(0,1,1):}]and A^(-1)=1/k` adj A, then k is

Options

A. 7
B. -7
C. `1`//7`
D. 11

Explanation

To find the value of \( k \) in the given problem, we need to compute the determinant of the matrix \( A \) and then use the relationship that \( A^{-1} = \frac{1}{k} \text{adj}(A) \).

### Step-by-Step Solution:

1. **Write down the matrix \( A \)**:
\[ A = \begin{pmatrix} 3 & 2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 \end{pmatrix} \]

2. **Calculate the determinant of matrix \( A \)**:
The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

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

Plugging in these values:
\[ \text{det}(A) = 3(2 \cdot 1 - (-1) \cdot 1) - 2(1 \cdot 1 - (-1) \cdot 0) + 4(1 \cdot 1 - 2 \cdot 0) \]

Simplifying each term:
- First term: \( 3(2 + 1) = 3 \cdot 3 = 9 \)
- Second term: \( -2(1 - 0) = -2 \cdot 1 = -2 \)
- Third term: \( 4(1 - 0) = 4 \cdot 1 = 4 \)

Now, combine these results:
\[ \text{det}(A) = 9 - 2 + 4 = 11 \]

3. **Relate \( k \) to the determinant**:
From the problem, we know that:
\[ A^{-1} = \frac{1}{k} \text{adj}(A) \]
and we also know that:
\[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \]
Therefore, we can conclude that:
\[ k = \text{det}(A) \]

4. **Final value of \( k \)**:
Since we calculated \( \text{det}(A) = 11 \), we have:
\[ k = 11 \]

### Conclusion:
The value of \( k \) is \( 11 \).