If `A= [(3,1),(4,0)], B= [(1,-2),(2,3)] and 3A- 5B + 2X= [(4,3),(0,1)]`, find the matrix X.

Answer

To find the matrix \( X \) given the equation \( 3A - 5B + 2X = \begin{pmatrix} 4 & 3 \\ 0 & 1 \end{pmatrix} \), we will follow these steps:

### Step 1: Write down the matrices
Given:
\[ A = \begin{pmatrix} 3 & 1 \\ 4 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & -2 \\ 2 & 3 \end{pmatrix} \]

### Step 2: Calculate \( 3A \)
To find \( 3A \), we multiply each element of matrix \( A \) by 3:
\[ 3A = 3 \times \begin{pmatrix} 3 & 1 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 3 \times 3 & 3 \times 1 \\ 3 \times 4 & 3 \times 0 \end{pmatrix} = \begin{pmatrix} 9 & 3 \\ 12 & 0 \end{pmatrix} \]

### Step 3: Calculate \( 5B \)
Next, we find \( 5B \) by multiplying each element of matrix \( B \) by 5:
\[ 5B = 5 \times \begin{pmatrix} 1 & -2 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 5 \times 1 & 5 \times -2 \\ 5 \times 2 & 5 \times 3 \end{pmatrix} = \begin{pmatrix} 5 & -10 \\ 10 & 15 \end{pmatrix} \]

### Step 4: Substitute \( 3A \) and \( 5B \) into the equation
Now substitute \( 3A \) and \( 5B \) into the equation:
\[ 3A - 5B + 2X = \begin{pmatrix} 9 & 3 \\ 12 & 0 \end{pmatrix} - \begin{pmatrix} 5 & -10 \\ 10 & 15 \end{pmatrix} + 2X = \begin{pmatrix} 4 & 3 \\ 0 & 1 \end{pmatrix} \]

### Step 5: Perform the subtraction
Now we perform the subtraction \( 3A - 5B \):
\[ \begin{pmatrix} 9 & 3 \\ 12 & 0 \end{pmatrix} - \begin{pmatrix} 5 & -10 \\ 10 & 15 \end{pmatrix} = \begin{pmatrix} 9 - 5 & 3 - (-10) \\ 12 - 10 & 0 - 15 \end{pmatrix} = \begin{pmatrix} 4 & 13 \\ 2 & -15 \end{pmatrix} \]

### Step 6: Rearrange the equation to isolate \( 2X \)
Now we have:
\[ \begin{pmatrix} 4 & 13 \\ 2 & -15 \end{pmatrix} + 2X = \begin{pmatrix} 4 & 3 \\ 0 & 1 \end{pmatrix} \]
Subtract \( \begin{pmatrix} 4 & 13 \\ 2 & -15 \end{pmatrix} \) from both sides:
\[ 2X = \begin{pmatrix} 4 & 3 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} 4 & 13 \\ 2 & -15 \end{pmatrix} = \begin{pmatrix} 4 - 4 & 3 - 13 \\ 0 - 2 & 1 - (-15) \end{pmatrix} = \begin{pmatrix} 0 & -10 \\ -2 & 16 \end{pmatrix} \]

### Step 7: Divide by 2 to find \( X \)
Finally, divide each element of \( 2X \) by 2 to find \( X \):
\[ X = \frac{1}{2} \begin{pmatrix} 0 & -10 \\ -2 & 16 \end{pmatrix} = \begin{pmatrix} 0 & -5 \\ -1 & 8 \end{pmatrix} \]

### Final Answer
Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} 0 & -5 \\ -1 & 8 \end{pmatrix} \]