If `[{:(,1,4),(,-2,3):}] +2M=3 [{:(,3,2),(,0,-3):}]` find the matrix M.

Answer

To solve the equation given in the question, we need to find the matrix \( M \) from the equation:

\[ \begin{pmatrix} 1 & 4 \\ -2 & 3 \end{pmatrix} + 2M = 3 \begin{pmatrix} 3 & 2 \\ 0 & -3 \end{pmatrix} \]

### Step 1: Multiply the matrix on the right side by 3
We start by calculating the right-hand side of the equation:

\[ 3 \begin{pmatrix} 3 & 2 \\ 0 & -3 \end{pmatrix} = \begin{pmatrix} 3 \times 3 & 3 \times 2 \\ 3 \times 0 & 3 \times -3 \end{pmatrix} = \begin{pmatrix} 9 & 6 \\ 0 & -9 \end{pmatrix} \]

### Step 2: Rewrite the equation
Now, we can rewrite the equation as:

\[ \begin{pmatrix} 1 & 4 \\ -2 & 3 \end{pmatrix} + 2M = \begin{pmatrix} 9 & 6 \\ 0 & -9 \end{pmatrix} \]

### Step 3: Move the matrix on the left side to the right side
Next, we subtract the matrix on the left side from both sides:

\[ 2M = \begin{pmatrix} 9 & 6 \\ 0 & -9 \end{pmatrix} - \begin{pmatrix} 1 & 4 \\ -2 & 3 \end{pmatrix} \]

### Step 4: Perform the subtraction
Now, we perform the matrix subtraction:

\[ \begin{pmatrix} 9 - 1 & 6 - 4 \\ 0 - (-2) & -9 - 3 \end{pmatrix} = \begin{pmatrix} 8 & 2 \\ 2 & -12 \end{pmatrix} \]

### Step 5: Divide by 2 to isolate M
Now, we divide the resulting matrix by 2 to solve for \( M \):

\[ M = \frac{1}{2} \begin{pmatrix} 8 & 2 \\ 2 & -12 \end{pmatrix} = \begin{pmatrix} \frac{8}{2} & \frac{2}{2} \\ \frac{2}{2} & \frac{-12}{2} \end{pmatrix} = \begin{pmatrix} 4 & 1 \\ 1 & -6 \end{pmatrix} \]

### Final Result
Thus, the matrix \( M \) is:

\[ M = \begin{pmatrix} 4 & 1 \\ 1 & -6 \end{pmatrix} \]

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