If `[(I,0),(3,-i)]+X = [(I,2),(3,4+i)]` -X then X is equal to

Options

A. `[(0,-1),(3,i)]`
B. `[(0,1),(0,2+i)]`
C. `[(1,0),(0,2-i)]`
D. `[(I,2),(0,2+i)]`

Answer

To solve the equation \(\begin{pmatrix} I & 0 \\ 3 & -i \end{pmatrix} + X = \begin{pmatrix} I & 2 \\ 3 & 4+i \end{pmatrix} - X\), we will follow these steps:

### Step 1: Move all matrices to one side and all variables to the other side
We start with the equation:
\[ \begin{pmatrix} I & 0 \\ 3 & -i \end{pmatrix} + X = \begin{pmatrix} I & 2 \\ 3 & 4+i \end{pmatrix} - X \]
We can add \(X\) to both sides and subtract the matrix \(\begin{pmatrix} I & 0 \\ 3 & -i \end{pmatrix}\) from both sides:
\[ X + X = \begin{pmatrix} I & 2 \\ 3 & 4+i \end{pmatrix} - \begin{pmatrix} I & 0 \\ 3 & -i \end{pmatrix} \]
This simplifies to:
\[ 2X = \begin{pmatrix} I & 2 \\ 3 & 4+i \end{pmatrix} - \begin{pmatrix} I & 0 \\ 3 & -i \end{pmatrix} \]

### Step 2: Perform the matrix subtraction on the right-hand side
Now we subtract the matrices on the right-hand side:
\[ \begin{pmatrix} I & 2 \\ 3 & 4+i \end{pmatrix} - \begin{pmatrix} I & 0 \\ 3 & -i \end{pmatrix} = \begin{pmatrix} I - I & 2 - 0 \\ 3 - 3 & (4+i) - (-i) \end{pmatrix} \]
Calculating each element:
- First element: \(I - I = 0\)
- Second element: \(2 - 0 = 2\)
- Third element: \(3 - 3 = 0\)
- Fourth element: \((4+i) - (-i) = 4 + i + i = 4 + 2i\)

Thus, we have:
\[ 2X = \begin{pmatrix} 0 & 2 \\ 0 & 4 + 2i \end{pmatrix} \]

### Step 3: Solve for \(X\)
To find \(X\), we divide both sides by 2:
\[ X = \frac{1}{2} \begin{pmatrix} 0 & 2 \\ 0 & 4 + 2i \end{pmatrix} \]
This results in:
\[ X = \begin{pmatrix} 0 & 1 \\ 0 & 2 + i \end{pmatrix} \]

### Final Answer
Thus, the value of \(X\) is:
\[ X = \begin{pmatrix} 0 & 1 \\ 0 & 2 + i \end{pmatrix} \]