Let `A=[{:(1,0),(0,-1):}] and B=[{:(1,x),(0,1):}]`. If AB=BA then what is the value of x ?

Options

A. `-1`
B. 0
C. 1
D. Any real number

Answer

To solve the problem where we need to find the value of \( x \) such that \( AB = BA \) for the given matrices \( A \) and \( B \), we will follow these steps:

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

### Step 2: Calculate the product \( AB \)
To find \( AB \), we perform the matrix multiplication:
\[ AB = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} \]
Calculating each element:
- First row, first column: \( 1 \cdot 1 + 0 \cdot 0 = 1 \)
- First row, second column: \( 1 \cdot x + 0 \cdot 1 = x \)
- Second row, first column: \( 0 \cdot 1 + (-1) \cdot 0 = 0 \)
- Second row, second column: \( 0 \cdot x + (-1) \cdot 1 = -1 \)

Thus,
\[ AB = \begin{pmatrix} 1 & x \\ 0 & -1 \end{pmatrix} \]

### Step 3: Calculate the product \( BA \)
Now, we calculate \( BA \):
\[ BA = \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]
Calculating each element:
- First row, first column: \( 1 \cdot 1 + x \cdot 0 = 1 \)
- First row, second column: \( 1 \cdot 0 + x \cdot (-1) = -x \)
- Second row, first column: \( 0 \cdot 1 + 1 \cdot 0 = 0 \)
- Second row, second column: \( 0 \cdot 0 + 1 \cdot (-1) = -1 \)

Thus,
\[ BA = \begin{pmatrix} 1 & -x \\ 0 & -1 \end{pmatrix} \]

### Step 4: Set \( AB \) equal to \( BA \)
Since we know \( AB = BA \), we set the two matrices equal to each other:
\[ \begin{pmatrix} 1 & x \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & -x \\ 0 & -1 \end{pmatrix} \]

### Step 5: Compare corresponding elements
From the matrices, we can compare the elements:
1. From the first row, second column: \( x = -x \)

### Step 6: Solve for \( x \)
To solve the equation \( x = -x \):
\[ x + x = 0 \implies 2x = 0 \implies x = 0 \]

### Conclusion
The value of \( x \) is:
\[ \boxed{0} \]