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

Options

A. `[(1,0),(0,-1)]`
B. `[(-1,0),(0,-1)]`
C. `[(1,0),(0,1)]`
D. `[(-1,0),(0,1)]`

Answer

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

### Step 1: Write down the matrix \( A \)
We have:
\[ A = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \]

### Step 2: Set up the multiplication for \( A^2 \)
We need to calculate \( A^2 \), which is \( A \times A \):
\[ A^2 = A \times A = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \times \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \]

### Step 3: Perform the multiplication
To multiply two matrices, we take the dot product of the rows of the first matrix with the columns of the second matrix.

**Calculating the first row and first column:**
\[ \text{Element (1,1)}: i \cdot i + 0 \cdot 0 = i^2 + 0 = i^2 \]

**Calculating the first row and second column:**
\[ \text{Element (1,2)}: i \cdot 0 + 0 \cdot i = 0 + 0 = 0 \]

**Calculating the second row and first column:**
\[ \text{Element (2,1)}: 0 \cdot i + i \cdot 0 = 0 + 0 = 0 \]

**Calculating the second row and second column:**
\[ \text{Element (2,2)}: 0 \cdot 0 + i \cdot i = 0 + i^2 = i^2 \]

### Step 4: Combine the results into the resulting matrix
Putting all the calculated elements together, we get:
\[ A^2 = \begin{pmatrix} i^2 & 0 \\ 0 & i^2 \end{pmatrix} \]

### Step 5: Substitute \( i^2 \) with its value
Since \( i^2 = -1 \), we substitute:
\[ A^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]

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