If A = `[(0,-i),(i,0)] B = [(1,0),(0,-1)]` then A B+ BA is

Options

A. null matrix
B. unit matrix
C. invertible matrix
D. none of these

Answer

To solve the problem, we need to find the sum of the products of matrices A and B, specifically \( AB + BA \).

Given:
\[ A = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \]
\[ B = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]

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

To find \( AB \), we multiply matrix A by matrix B:

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

Calculating each element of the resulting matrix:

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

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

### Step 2: Calculate \( BA \)

Next, we calculate \( BA \) by multiplying matrix B by matrix A:

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

Calculating each element of the resulting matrix:

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

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

### Step 3: Calculate \( AB + BA \)

Now, we add the matrices \( AB \) and \( BA \):

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

Calculating each element of the resulting matrix:

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

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

### Final Answer:
\[ AB + BA = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

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