Evaluate `|{:(1,2,3),(3,-1,1),(2,3,5):}|`

Options

A. `-1`
B. 7
C. `51`
D. `-4`

Answer

To evaluate the determinant of the matrix given by the rows (1, 2, 3), (3, -1, 1), and (2, 3, 5), we can follow these steps:

### Step 1: Write down the matrix
We start by writing the matrix corresponding to the given rows:
\[ \begin{bmatrix} 1 & 2 & 3 \\ 3 & -1 & 1 \\ 2 & 3 & 5 \end{bmatrix} \]

### Step 2: Use the determinant formula for a 3x3 matrix
The determinant of a 3x3 matrix
\[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]
is calculated using the formula:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

### Step 3: Identify the elements
From our matrix, we have:
- \( a = 1, b = 2, c = 3 \)
- \( d = 3, e = -1, f = 1 \)
- \( g = 2, h = 3, i = 5 \)

### Step 4: Substitute into the determinant formula
Now we substitute these values into the determinant formula:
\[ |A| = 1((-1)(5) - (1)(3)) - 2((3)(5) - (1)(2)) + 3((3)(3) - (-1)(2)) \]

### Step 5: Calculate each term
Now we calculate each term:
1. First term:
\[ 1((-1)(5) - (1)(3)) = 1(-5 - 3) = 1(-8) = -8 \]
2. Second term:
\[ -2((3)(5) - (1)(2)) = -2(15 - 2) = -2(13) = -26 \]
3. Third term:
\[ 3((3)(3) - (-1)(2)) = 3(9 + 2) = 3(11) = 33 \]

### Step 6: Combine the results
Now we combine all the results:
\[ |A| = -8 - 26 + 33 \]
Calculating this gives:
\[ |A| = -34 + 33 = -1 \]

### Final Answer
Thus, the value of the determinant is:
\[ \boxed{-1} \]