If `A= [(0,1),(1,0)] and B= [(0,-x),(x,0)]`, then

Options

A. AB=BA
B. `AB + BA= 1`
C. `AB+BA= 0`
D. None of these

Answer

To solve the problem, we need to find the product of the matrices \( A \) and \( B \), as well as the product of \( B \) and \( A \), and then check the condition \( AB + BA = 0 \).

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

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

Using the formula for matrix multiplication, the entry in the first row and first column of \( AB \) is calculated as:
\[ AB_{11} = A_{11}B_{11} + A_{12}B_{21} = 0 \cdot 0 + 1 \cdot x = x \]

The entry in the first row and second column is:
\[ AB_{12} = A_{11}B_{12} + A_{12}B_{22} = 0 \cdot (-x) + 1 \cdot 0 = 0 \]

The entry in the second row and first column is:
\[ AB_{21} = A_{21}B_{11} + A_{22}B_{21} = 1 \cdot 0 + 0 \cdot x = 0 \]

The entry in the second row and second column is:
\[ AB_{22} = A_{21}B_{12} + A_{22}B_{22} = 1 \cdot (-x) + 0 \cdot 0 = -x \]

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

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

Now, we calculate \( BA \):
The entry in the first row and first column of \( BA \) is:
\[ BA_{11} = B_{11}A_{11} + B_{12}A_{21} = 0 \cdot 0 + (-x) \cdot 1 = -x \]

The entry in the first row and second column is:
\[ BA_{12} = B_{11}A_{12} + B_{12}A_{22} = 0 \cdot 1 + (-x) \cdot 0 = 0 \]

The entry in the second row and first column is:
\[ BA_{21} = B_{21}A_{11} + B_{22}A_{21} = x \cdot 0 + 0 \cdot 1 = 0 \]

The entry in the second row and second column is:
\[ BA_{22} = B_{21}A_{12} + B_{22}A_{22} = x \cdot 1 + 0 \cdot 0 = x \]

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

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

Now, we add \( AB \) and \( BA \):
\[ AB + BA = \begin{pmatrix} x & 0 \\ 0 & -x \end{pmatrix} + \begin{pmatrix} -x & 0 \\ 0 & x \end{pmatrix} = \begin{pmatrix} x - x & 0 + 0 \\ 0 + 0 & -x + x \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Conclusion

Thus, we have shown that:
\[ AB + BA = 0 \]
This confirms that the condition \( AB + BA = 0 \) is true.