The matrix A =`[{:(-3,x),(4,2//3):}]` does not have an inverse. Find the value of x.

Options

A. `-2/3`
B. `-1/2`
C. 0
D. `3/4`

Explanation

To find the value of \( x \) such that the matrix

\[ A = \begin{pmatrix} -3 & x \\ 4 & \frac{2}{3} \end{pmatrix} \]

does not have an inverse, we need to determine when the determinant of the matrix is equal to zero. A matrix does not have an inverse if its determinant is zero.

### Step-by-Step Solution:

1. **Calculate the Determinant of Matrix A**:
The determinant of a 2x2 matrix

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

is given by the formula:

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

For our matrix \( A \):

- \( a = -3 \)
- \( b = x \)
- \( c = 4 \)
- \( d = \frac{2}{3} \)

Therefore, the determinant of \( A \) is:

\[ \text{det}(A) = (-3) \cdot \left(\frac{2}{3}\right) - (4) \cdot (x) \]

Simplifying this gives:

\[ \text{det}(A) = -2 - 4x \]

2. **Set the Determinant Equal to Zero**:
To find the value of \( x \) for which the matrix does not have an inverse, we set the determinant equal to zero:

\[ -2 - 4x = 0 \]

3. **Solve for x**:
Rearranging the equation gives:

\[ -4x = 2 \]

Dividing both sides by -4:

\[ x = -\frac{2}{4} = -\frac{1}{2} \]

### Final Answer:
The value of \( x \) such that the matrix \( A \) does not have an inverse is:

\[ \boxed{-\frac{1}{2}} \]