Solve for matrics A and B , where
` 2A + B = [{:( 3,-4),(2,7) :}] and A - 2B =[{:( 4,3),(1,1):}]`

Answer

To solve for matrices A and B given the equations:

1. \( 2A + B = \begin{pmatrix} 3 & -4 \\ 2 & 7 \end{pmatrix} \) (Equation 1)
2. \( A - 2B = \begin{pmatrix} 4 & 3 \\ 1 & 1 \end{pmatrix} \) (Equation 2)

We will follow these steps:

### Step 1: Multiply Equation 1 by 2
Multiply the entire Equation 1 by 2 to eliminate B later.

\[ 2(2A + B) = 2 \begin{pmatrix} 3 & -4 \\ 2 & 7 \end{pmatrix} \]

This simplifies to:

\[ 4A + 2B = \begin{pmatrix} 6 & -8 \\ 4 & 14 \end{pmatrix} \quad \text{(Equation 3)} \]

**Hint:** When multiplying a matrix by a scalar, multiply each element of the matrix by that scalar.

### Step 2: Add Equation 2 and Equation 3
Now, we will add Equation 2 and Equation 3:

\[ (4A + 2B) + (A - 2B) = \begin{pmatrix} 6 & -8 \\ 4 & 14 \end{pmatrix} + \begin{pmatrix} 4 & 3 \\ 1 & 1 \end{pmatrix} \]

This simplifies to:

\[ 5A = \begin{pmatrix} 6 + 4 & -8 + 3 \\ 4 + 1 & 14 + 1 \end{pmatrix} \]

Calculating the right side gives:

\[ 5A = \begin{pmatrix} 10 & -5 \\ 5 & 15 \end{pmatrix} \]

**Hint:** When adding two matrices, add the corresponding elements together.

### Step 3: Solve for A
Now, divide both sides by 5 to find A:

\[ A = \frac{1}{5} \begin{pmatrix} 10 & -5 \\ 5 & 15 \end{pmatrix} \]

Calculating this gives:

\[ A = \begin{pmatrix} \frac{10}{5} & \frac{-5}{5} \\ \frac{5}{5} & \frac{15}{5} \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix} \]

**Hint:** To divide a matrix by a scalar, divide each element of the matrix by that scalar.

### Step 4: Substitute A into Equation 1 to find B
Now substitute A back into Equation 1 to find B:

\[ 2A + B = \begin{pmatrix} 3 & -4 \\ 2 & 7 \end{pmatrix} \]

Substituting A:

\[ 2 \begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix} + B = \begin{pmatrix} 3 & -4 \\ 2 & 7 \end{pmatrix} \]

Calculating \(2A\):

\[ \begin{pmatrix} 4 & -2 \\ 2 & 6 \end{pmatrix} + B = \begin{pmatrix} 3 & -4 \\ 2 & 7 \end{pmatrix} \]

### Step 5: Solve for B
Now, isolate B:

\[ B = \begin{pmatrix} 3 & -4 \\ 2 & 7 \end{pmatrix} - \begin{pmatrix} 4 & -2 \\ 2 & 6 \end{pmatrix} \]

Calculating this gives:

\[ B = \begin{pmatrix} 3 - 4 & -4 - (-2) \\ 2 - 2 & 7 - 6 \end{pmatrix} = \begin{pmatrix} -1 & -2 \\ 0 & 1 \end{pmatrix} \]

### Final Result
Thus, the matrices A and B are:

\[ A = \begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & -2 \\ 0 & 1 \end{pmatrix} \]

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