If `A=[(2,-2),(4,2),(-5,1)],B=[(8,0),(4,-2),(3,6)]`, find matrix 'C', such that `2A+3C=5B`

Answer

To find the matrix \( C \) such that \( 2A + 3C = 5B \), we will follow these steps:

### Step 1: Write down the matrices \( A \) and \( B \)

Given:
\[ A = \begin{pmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{pmatrix} \]

### Step 2: Calculate \( 2A \)

To find \( 2A \), we multiply each element of matrix \( A \) by 2:
\[ 2A = 2 \times \begin{pmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{pmatrix} = \begin{pmatrix} 4 & -4 \\ 8 & 4 \\ -10 & 2 \end{pmatrix} \]

### Step 3: Calculate \( 5B \)

Next, we calculate \( 5B \) by multiplying each element of matrix \( B \) by 5:
\[ 5B = 5 \times \begin{pmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{pmatrix} = \begin{pmatrix} 40 & 0 \\ 20 & -10 \\ 15 & 30 \end{pmatrix} \]

### Step 4: Set up the equation \( 3C = 5B - 2A \)

Now we substitute \( 2A \) and \( 5B \) into the equation:
\[ 3C = 5B - 2A = \begin{pmatrix} 40 & 0 \\ 20 & -10 \\ 15 & 30 \end{pmatrix} - \begin{pmatrix} 4 & -4 \\ 8 & 4 \\ -10 & 2 \end{pmatrix} \]

### Step 5: Perform the subtraction

Now we perform the subtraction element-wise:
\[ 3C = \begin{pmatrix} 40 - 4 & 0 - (-4) \\ 20 - 8 & -10 - 4 \\ 15 - (-10) & 30 - 2 \end{pmatrix} = \begin{pmatrix} 36 & 4 \\ 12 & -14 \\ 25 & 28 \end{pmatrix} \]

### Step 6: Solve for \( C \)

To find \( C \), we divide each element of \( 3C \) by 3:
\[ C = \frac{1}{3} \begin{pmatrix} 36 & 4 \\ 12 & -14 \\ 25 & 28 \end{pmatrix} = \begin{pmatrix} 12 & \frac{4}{3} \\ 4 & -\frac{14}{3} \\ \frac{25}{3} & \frac{28}{3} \end{pmatrix} \]

### Final Answer

Thus, the matrix \( C \) is:
\[ C = \begin{pmatrix} 12 & \frac{4}{3} \\ 4 & -\frac{14}{3} \\ \frac{25}{3} & \frac{28}{3} \end{pmatrix} \]