If `A =({:(i,0),(0,-i):}),B=({:(0,-1),(1,0):}),C=({:(0,i),(i,0):})` wher `i = sqrt(-1)`, then which one of the following is correct ?

Options

A. AB = -C
B. AB = C
C. `A^(2)=B^(2)=C^(2)=I`, where I is the identity matrix
D. `BA ne C`

Explanation

To solve the problem, we need to compute the product of matrices \( A \) and \( B \) and then compare it with matrix \( C \) to find the correct relationship.

Given:
\[ A = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \quad B = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \]
where \( i = \sqrt{-1} \).

### Step 1: Calculate the product \( AB \)

To find \( AB \), we multiply the matrices \( A \) and \( B \):

\[ AB = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

Using the formula for matrix multiplication, we calculate each element of the resulting matrix:

- First row, first column:
\[ (i \cdot 0) + (0 \cdot 1) = 0 \]

- First row, second column:
\[ (i \cdot -1) + (0 \cdot 0) = -i \]

- Second row, first column:
\[ (0 \cdot 0) + (-i \cdot 1) = -i \]

- Second row, second column:
\[ (0 \cdot -1) + (-i \cdot 0) = 0 \]

Thus, we have:
\[ AB = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix} \]

### Step 2: Compare \( AB \) with \( C \)

Now, we compare \( AB \) with \( C \):
\[ C = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \]

Notice that:
\[ AB = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix} = -\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} = -C \]

### Conclusion

From our calculations, we find that:
\[ AB = -C \]
This means that \( C = -AB \).

### Final Answer

The correct relationship is:
\[ C = -AB \]