If `A=[[i, 0], [0, (i)/(2)]]`, then `A^(-1)=`

Options

A. `[[i, 0], [0, (i)/(2)]]`
B. `[[i, 0], [0, 2i]]`
C. `[[-i, 0], [0, -2i]]`
D. `[[0, i], [2i, 0]]`

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} i & 0 \\ 0 & \frac{i}{2} \end{bmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of \( A \)

The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is calculated using the formula:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

- \( a = i \)
- \( b = 0 \)
- \( c = 0 \)
- \( d = \frac{i}{2} \)

Thus, the determinant is:

\[ \text{det}(A) = i \cdot \frac{i}{2} - 0 \cdot 0 = \frac{i^2}{2} = \frac{-1}{2} \]

### Step 2: Calculate the Adjoint of \( A \)

The adjoint of a 2x2 matrix is given by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. For our matrix \( A \):

\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} \frac{i}{2} & 0 \\ 0 & i \end{bmatrix} \]

### Step 3: Calculate the Inverse of \( A \)

The inverse of a matrix is given by the formula:

\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]

Substituting the values we found:

\[ A^{-1} = \frac{1}{\left(-\frac{1}{2}\right)} \cdot \begin{bmatrix} \frac{i}{2} & 0 \\ 0 & i \end{bmatrix} \]

This simplifies to:

\[ A^{-1} = -2 \cdot \begin{bmatrix} \frac{i}{2} & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} -i & 0 \\ 0 & -2i \end{bmatrix} \]

### Final Answer

Thus, the inverse of the matrix \( A \) is:

\[ A^{-1} = \begin{bmatrix} -i & 0 \\ 0 & -2i \end{bmatrix} \]

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