matrix `A=({:(-i,0),(0,-i):})` then `A^2` =

Options

A. 2I
B. `-I`
C. I
D. None of these

Answer

To find \( A^2 \) for the matrix \( A = \begin{pmatrix} -i & 0 \\ 0 & -i \end{pmatrix} \), we will perform matrix multiplication.

### Step 1: Write down the matrix multiplication
We need to compute \( A^2 = A \times A \):
\[ A^2 = \begin{pmatrix} -i & 0 \\ 0 & -i \end{pmatrix} \times \begin{pmatrix} -i & 0 \\ 0 & -i \end{pmatrix} \]

### Step 2: Perform the multiplication
Using the rules of matrix multiplication, we calculate each element of the resulting matrix:

1. **First row, first column**:
\[ (-i) \times (-i) + 0 \times 0 = i^2 = -1 \]

2. **First row, second column**:
\[ (-i) \times 0 + 0 \times (-i) = 0 + 0 = 0 \]

3. **Second row, first column**:
\[ 0 \times (-i) + (-i) \times 0 = 0 + 0 = 0 \]

4. **Second row, second column**:
\[ 0 \times 0 + (-i) \times (-i) = 0 + i^2 = -1 \]

### Step 3: Combine the results
Putting all the calculated elements together, we get:
\[ A^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]

### Final Result
Thus, the final result is:
\[ A^2 = -I = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]