If ` P=[(6,-2),(4,-6):}] and Q = [{:(5,3),(2,0):}]` find the matrix M such that ` 2Q - 3P - 3M =0 `

Answer

To solve the problem, we need to find the matrix \( M \) such that the equation \( 2Q - 3P - 3M = 0 \) holds true.

### Step-by-Step Solution:

1. **Identify the matrices**:
\[ P = \begin{pmatrix} 6 & -2 \\ 4 & -6 \end{pmatrix}, \quad Q = \begin{pmatrix} 5 & 3 \\ 2 & 0 \end{pmatrix} \]

2. **Calculate \( 2Q \)**:
\[ 2Q = 2 \times \begin{pmatrix} 5 & 3 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} 2 \times 5 & 2 \times 3 \\ 2 \times 2 & 2 \times 0 \end{pmatrix} = \begin{pmatrix} 10 & 6 \\ 4 & 0 \end{pmatrix} \]

3. **Calculate \( -3P \)**:
\[ -3P = -3 \times \begin{pmatrix} 6 & -2 \\ 4 & -6 \end{pmatrix} = \begin{pmatrix} -3 \times 6 & -3 \times -2 \\ -3 \times 4 & -3 \times -6 \end{pmatrix} = \begin{pmatrix} -18 & 6 \\ -12 & 18 \end{pmatrix} \]

4. **Combine \( 2Q \) and \( -3P \)**:
\[ 2Q - 3P = \begin{pmatrix} 10 & 6 \\ 4 & 0 \end{pmatrix} + \begin{pmatrix} -18 & 6 \\ -12 & 18 \end{pmatrix} = \begin{pmatrix} 10 - 18 & 6 + 6 \\ 4 - 12 & 0 + 18 \end{pmatrix} = \begin{pmatrix} -8 & 12 \\ -8 & 18 \end{pmatrix} \]

5. **Set up the equation**:
\[ 2Q - 3P - 3M = 0 \implies 3M = 2Q - 3P \]
Thus, we have:
\[ 3M = \begin{pmatrix} -8 & 12 \\ -8 & 18 \end{pmatrix} \]

6. **Solve for \( M \)**:
\[ M = \frac{1}{3} \begin{pmatrix} -8 & 12 \\ -8 & 18 \end{pmatrix} = \begin{pmatrix} \frac{-8}{3} & \frac{12}{3} \\ \frac{-8}{3} & \frac{18}{3} \end{pmatrix} = \begin{pmatrix} -\frac{8}{3} & 4 \\ -\frac{8}{3} & 6 \end{pmatrix} \]

### Final Result:
\[ M = \begin{pmatrix} -\frac{8}{3} & 4 \\ -\frac{8}{3} & 6 \end{pmatrix} \]