If `(A-2B)=[(1,-2),(3,0)] and (2A-3B)=[(-2,2),(3,-3)]` then B=?

Options

A. `[(6,-4),(-3,3)]`
B. `[(-4,6),(-3,-3)]`
C. `[(4,-6),(3,-3)]`
D. none of these

Explanation

To solve the problem, we are given two equations involving matrices \( A \) and \( B \):

1. \( A - 2B = \begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix} \)
2. \( 2A - 3B = \begin{pmatrix} -2 & 2 \\ 3 & -3 \end{pmatrix} \)

We need to find the matrix \( B \).

### Step 1: Express \( A \) in terms of \( B \)
From the first equation, we can express \( A \) as:
\[ A = 2B + \begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix} \]

### Step 2: Substitute \( A \) into the second equation
Now, substitute this expression for \( A \) into the second equation:
\[ 2(2B + \begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix}) - 3B = \begin{pmatrix} -2 & 2 \\ 3 & -3 \end{pmatrix} \]

### Step 3: Simplify the equation
Expanding the left side:
\[ 4B + \begin{pmatrix} 2 & -4 \\ 6 & 0 \end{pmatrix} - 3B = \begin{pmatrix} -2 & 2 \\ 3 & -3 \end{pmatrix} \]
This simplifies to:
\[ (4B - 3B) + \begin{pmatrix} 2 & -4 \\ 6 & 0 \end{pmatrix} = \begin{pmatrix} -2 & 2 \\ 3 & -3 \end{pmatrix} \]
\[ B + \begin{pmatrix} 2 & -4 \\ 6 & 0 \end{pmatrix} = \begin{pmatrix} -2 & 2 \\ 3 & -3 \end{pmatrix} \]

### Step 4: Isolate \( B \)
Now, isolate \( B \):
\[ B = \begin{pmatrix} -2 & 2 \\ 3 & -3 \end{pmatrix} - \begin{pmatrix} 2 & -4 \\ 6 & 0 \end{pmatrix} \]

### Step 5: Perform the matrix subtraction
Subtract the matrices:
\[ B = \begin{pmatrix} -2 - 2 & 2 - (-4) \\ 3 - 6 & -3 - 0 \end{pmatrix} \]
\[ B = \begin{pmatrix} -4 & 6 \\ -3 & -3 \end{pmatrix} \]

### Final Result
Thus, the matrix \( B \) is:
\[ B = \begin{pmatrix} -4 & 6 \\ -3 & -3 \end{pmatrix} \]