The value of `|(11,12,13),(12,13,14),(13,14,15)|`, is

Options

A. 1
B. 0
C. `-1`
D. 67

Answer

To find the value of the determinant \( |(11, 12, 13), (12, 13, 14), (13, 14, 15)| \), we can follow these steps:

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} 11 & 12 & 13 \\ 12 & 13 & 14 \\ 13 & 14 & 15 \end{vmatrix} \]

### Step 2: Apply Row Operations
We can simplify the determinant by performing row operations. Let's subtract the first row from the second row and the first row from the third row.

- For Row 2: \( R_2 \rightarrow R_2 - R_1 \)
- For Row 3: \( R_3 \rightarrow R_3 - R_1 \)

This gives us:
\[ D = \begin{vmatrix} 11 & 12 & 13 \\ 12 - 11 & 13 - 12 & 14 - 13 \\ 13 - 11 & 14 - 12 & 15 - 13 \end{vmatrix} \]
Which simplifies to:
\[ D = \begin{vmatrix} 11 & 12 & 13 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \end{vmatrix} \]

### Step 3: Factor Out Common Terms
Now, we can factor out the common term from the third row:
\[ D = 2 \cdot \begin{vmatrix} 11 & 12 & 13 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} \]

### Step 4: Identify Identical Rows
Notice that the second and third rows are identical:
\[ \begin{vmatrix} 11 & 12 & 13 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} \]
Since two rows are identical, the value of the determinant is 0.

### Step 5: Final Calculation
Thus, we have:
\[ D = 2 \cdot 0 = 0 \]

### Conclusion
The value of the determinant \( |(11, 12, 13), (12, 13, 14), (13, 14, 15)| \) is \( 0 \).