Evaluate `|{:(2, 7, 24), (6, 16, 72), (5, 9, 60):}|`

Options

A. 0
B. 1024
C. 1
D. None of these

Answer

To evaluate the determinant \( | \begin{vmatrix} 2 & 7 & 24 \\ 6 & 16 & 72 \\ 5 & 9 & 60 \end{vmatrix} | \), we will follow these steps:

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} 2 & 7 & 24 \\ 6 & 16 & 72 \\ 5 & 9 & 60 \end{vmatrix} \]

### Step 2: Apply Column Operations
We can simplify the determinant by performing column operations. Let's perform the operation \( C_3 \leftarrow C_3 - 3C_1 \). This means we will subtract three times the first column from the third column.

Calculating \( C_3 - 3C_1 \):
- For the first row: \( 24 - 3 \times 2 = 24 - 6 = 18 \)
- For the second row: \( 72 - 3 \times 6 = 72 - 18 = 54 \)
- For the third row: \( 60 - 3 \times 5 = 60 - 15 = 45 \)

Now, the determinant becomes:
\[ D = \begin{vmatrix} 2 & 7 & 18 \\ 6 & 16 & 54 \\ 5 & 9 & 45 \end{vmatrix} \]

### Step 3: Apply Another Column Operation
Now, we can perform another operation \( C_2 \leftarrow C_2 - C_1 \):
Calculating \( C_2 - C_1 \):
- For the first row: \( 7 - 2 = 5 \)
- For the second row: \( 16 - 6 = 10 \)
- For the third row: \( 9 - 5 = 4 \)

Now, the determinant becomes:
\[ D = \begin{vmatrix} 2 & 5 & 18 \\ 6 & 10 & 54 \\ 5 & 4 & 45 \end{vmatrix} \]

### Step 4: Calculate the Determinant
Now, we can calculate the determinant using the formula:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Where:
- \( a = 2, b = 5, c = 18 \)
- \( d = 6, e = 10, f = 54 \)
- \( g = 5, h = 4, i = 45 \)

Calculating the individual parts:
1. \( ei - fh = 10 \times 45 - 54 \times 4 = 450 - 216 = 234 \)
2. \( di - fg = 6 \times 45 - 54 \times 5 = 270 - 270 = 0 \)
3. \( dh - eg = 6 \times 4 - 10 \times 5 = 24 - 50 = -26 \)

Putting these values into the determinant formula:
\[ D = 2(234) - 5(0) + 18(-26) = 468 + 0 - 468 = 0 \]

### Final Result
Thus, the value of the determinant is:
\[ \boxed{0} \]