If `[(2,-1),(2,0)]+2A=[(-3,5),(4,3)]` the matrix A equals

Options

A. `[(-5,6),(2,3)]`
B. `[(-5/2,3),(1,3/2)]`
C. `[(-5/2,6),(2,3)]`
D. `[(-5,8),(1,3)]`

Answer

To solve the equation \([(2,-1),(2,0)]+2A=[(-3,5),(4,3)]\) for the matrix \(A\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (2, -1) \quad (2, 0) + 2A = (-3, 5) \quad (4, 3) \]

### Step 2: Isolate \(2A\)
To isolate \(2A\), we can subtract the matrix \((2, -1) \quad (2, 0)\) from both sides:
\[ 2A = [(-3, 5) \quad (4, 3)] - [(2, -1) \quad (2, 0)] \]

### Step 3: Perform the subtraction
Now we perform the subtraction element-wise:
\[ 2A = \begin{pmatrix} -3 - 2 & 5 - (-1) \\ 4 - 2 & 3 - 0 \end{pmatrix} \]
Calculating each element:
- First element: \(-3 - 2 = -5\)
- Second element: \(5 - (-1) = 5 + 1 = 6\)
- Third element: \(4 - 2 = 2\)
- Fourth element: \(3 - 0 = 3\)

So we have:
\[ 2A = \begin{pmatrix} -5 & 6 \\ 2 & 3 \end{pmatrix} \]

### Step 4: Solve for \(A\)
To find \(A\), we divide each element of \(2A\) by 2:
\[ A = \frac{1}{2} \begin{pmatrix} -5 & 6 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} -\frac{5}{2} & 3 \\ 1 & \frac{3}{2} \end{pmatrix} \]

### Final Answer
Thus, the matrix \(A\) is:
\[ A = \begin{pmatrix} -\frac{5}{2} & 3 \\ 1 & \frac{3}{2} \end{pmatrix} \]

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