The value of the determinant `|(x,x+1),(x-1,x)|` is :

Options

A. 1
B. `-1`
C. 2
D. 0

Answer

To find the value of the determinant \( |(x, x+1), (x-1, x)| \), we will use the formula for the determinant of a 2x2 matrix. The determinant of a matrix

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

is calculated using the formula:

\[ \text{det} = ad - bc \]

### Step-by-Step Solution:

1. **Identify the elements of the matrix**:
The given matrix is

\[ \begin{pmatrix} x & x+1 \\ x-1 & x \end{pmatrix} \]

Here, \( a = x \), \( b = x + 1 \), \( c = x - 1 \), and \( d = x \).

2. **Apply the determinant formula**:
Using the formula \( \text{det} = ad - bc \):

\[ \text{det} = (x)(x) - (x + 1)(x - 1) \]

3. **Calculate \( ad \)**:
\[ ad = x^2 \]

4. **Calculate \( bc \)**:
First, expand \( (x + 1)(x - 1) \) using the difference of squares:

\[ (x + 1)(x - 1) = x^2 - 1 \]

5. **Substitute back into the determinant formula**:
Now substitute \( ad \) and \( bc \) back into the determinant formula:

\[ \text{det} = x^2 - (x^2 - 1) \]

6. **Simplify the expression**:
\[ \text{det} = x^2 - x^2 + 1 = 1 \]

### Final Answer:
The value of the determinant is \( 1 \).