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

Options

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

Answer

To find the value of the determinant \( |(x, x+1), (x-1, 1)| \), we can follow these steps:

### Step 1: Write down the determinant
The determinant of a 2x2 matrix is given by the formula:
\[ |A| = ad - bc \]
where \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \). In our case, we have:
\[ A = \begin{pmatrix} x & x+1 \\ x-1 & 1 \end{pmatrix} \]
Thus, \( a = x \), \( b = x + 1 \), \( c = x - 1 \), and \( d = 1 \).

### Step 2: Apply the determinant formula
Using the formula for the determinant, we compute:
\[ |A| = x \cdot 1 - (x + 1)(x - 1) \]

### Step 3: Simplify the expression
Now, we need to simplify the expression:
\[ |A| = x - (x + 1)(x - 1) \]
Next, we expand \( (x + 1)(x - 1) \):
\[ (x + 1)(x - 1) = x^2 - 1 \]
So, substituting this back into our determinant gives:
\[ |A| = x - (x^2 - 1) \]

### Step 4: Further simplify
Now, we can simplify this expression:
\[ |A| = x - x^2 + 1 \]
Rearranging the terms, we get:
\[ |A| = -x^2 + x + 1 \]

### Final Result
Thus, the value of the determinant is:
\[ |A| = -x^2 + x + 1 \]

---