If `A=[{:(,0,-1),(,4,-3):}], B=[{:(,-5),(,6):}] and 3A xx M=2B`, find matrix M.

Answer

To solve the problem, we need to find the matrix \( M \) given the equation \( 3A \times M = 2B \), where \( A = \begin{pmatrix} 0 & -1 \\ 4 & -3 \end{pmatrix} \) and \( B = \begin{pmatrix} -5 \\ 6 \end{pmatrix} \).

### Step-by-Step Solution:

1. **Write the given equation:**
\[ 3A \times M = 2B \]

2. **Substitute the matrices \( A \) and \( B \):**
\[ 3 \begin{pmatrix} 0 & -1 \\ 4 & -3 \end{pmatrix} \times M = 2 \begin{pmatrix} -5 \\ 6 \end{pmatrix} \]

3. **Calculate \( 2B \):**
\[ 2B = 2 \begin{pmatrix} -5 \\ 6 \end{pmatrix} = \begin{pmatrix} -10 \\ 12 \end{pmatrix} \]

4. **Calculate \( 3A \):**
\[ 3A = 3 \begin{pmatrix} 0 & -1 \\ 4 & -3 \end{pmatrix} = \begin{pmatrix} 0 & -3 \\ 12 & -9 \end{pmatrix} \]

5. **Let \( M \) be represented as:**
\[ M = \begin{pmatrix} X \\ Y \end{pmatrix} \]

6. **Set up the equation:**
\[ \begin{pmatrix} 0 & -3 \\ 12 & -9 \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} -10 \\ 12 \end{pmatrix} \]

7. **Perform the matrix multiplication:**
- For the first row:
\[ 0 \cdot X + (-3) \cdot Y = -3Y = -10 \implies Y = \frac{10}{3} \]
- For the second row:
\[ 12 \cdot X + (-9) \cdot Y = 12X - 9Y = 12 \]

8. **Substitute \( Y \) into the second equation:**
\[ 12X - 9 \left(\frac{10}{3}\right) = 12 \]
\[ 12X - 30 = 12 \implies 12X = 42 \implies X = \frac{42}{12} = \frac{7}{2} \]

9. **Write the final matrix \( M \):**
\[ M = \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} \frac{7}{2} \\ \frac{10}{3} \end{pmatrix} \]

### Final Answer:
\[ M = \begin{pmatrix} \frac{7}{2} \\ \frac{10}{3} \end{pmatrix} \]