If `A=[{:(1,0,-2),(2,-3,4):}]`, then the matrix X for which 2X + 3A = 0 holds true is

Options

A. `[{:(-(3)/(2),0,-3),(-3,-(9)/(2),-6):}]`
B. `[{:((3)/(2),0,-3),(3,-(9)/(2),-6):}]`
C. `[{:((3)/(2),0,3),(3,(9)/(2),6):}]`
D. `[{:(-(3)/(2),0,3),(-3,(9)/(2),-6):}]`

Explanation

To solve the problem, we need to find the matrix \( X \) such that \( 2X + 3A = 0 \), where \( A = \begin{pmatrix} 1 & 0 & -2 \\ 2 & -3 & 4 \end{pmatrix} \).

### Step-by-step Solution:

1. **Write the equation**: Start with the equation given in the problem:
\[ 2X + 3A = 0 \]

2. **Isolate \( 2X \)**: Rearranging the equation gives:
\[ 2X = -3A \]

3. **Substitute the value of \( A \)**: We know that \( A = \begin{pmatrix} 1 & 0 & -2 \\ 2 & -3 & 4 \end{pmatrix} \). Substitute this into the equation:
\[ 2X = -3 \begin{pmatrix} 1 & 0 & -2 \\ 2 & -3 & 4 \end{pmatrix} \]

4. **Multiply the matrix by -3**: Perform the multiplication:
\[ -3 \begin{pmatrix} 1 & 0 & -2 \\ 2 & -3 & 4 \end{pmatrix} = \begin{pmatrix} -3 & 0 & 6 \\ -6 & 9 & -12 \end{pmatrix} \]

5. **Set the equation for \( X \)**: Now we have:
\[ 2X = \begin{pmatrix} -3 & 0 & 6 \\ -6 & 9 & -12 \end{pmatrix} \]

6. **Divide by 2**: To find \( X \), divide both sides by 2:
\[ X = \frac{1}{2} \begin{pmatrix} -3 & 0 & 6 \\ -6 & 9 & -12 \end{pmatrix} = \begin{pmatrix} -\frac{3}{2} & 0 & 3 \\ -3 & \frac{9}{2} & -6 \end{pmatrix} \]

Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} -\frac{3}{2} & 0 & 3 \\ -3 & \frac{9}{2} & -6 \end{pmatrix} \]