Evaluate the following determinant: `|[1, 4, 9],[ 4, 9, 16],[ 9 ,16, 25]|`

Options

A. -6
B. 4
C. -8
D. 5

Explanation

To evaluate the determinant

\[ D = \begin{vmatrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \end{vmatrix} \]

we will use the method of cofactor expansion along the first row.

### Step 1: Expand the determinant along the first row
Using the formula for the determinant, we have:

\[ D = 1 \cdot \begin{vmatrix} 9 & 16 \\ 16 & 25 \end{vmatrix} - 4 \cdot \begin{vmatrix} 4 & 16 \\ 9 & 25 \end{vmatrix} + 9 \cdot \begin{vmatrix} 4 & 9 \\ 9 & 16 \end{vmatrix} \]

### Step 2: Calculate the 2x2 determinants
Now we will calculate each of the 2x2 determinants.

1. For the first determinant:

\[ \begin{vmatrix} 9 & 16 \\ 16 & 25 \end{vmatrix} = (9 \cdot 25) - (16 \cdot 16) = 225 - 256 = -31 \]

2. For the second determinant:

\[ \begin{vmatrix} 4 & 16 \\ 9 & 25 \end{vmatrix} = (4 \cdot 25) - (16 \cdot 9) = 100 - 144 = -44 \]

3. For the third determinant:

\[ \begin{vmatrix} 4 & 9 \\ 9 & 16 \end{vmatrix} = (4 \cdot 16) - (9 \cdot 9) = 64 - 81 = -17 \]

### Step 3: Substitute back into the determinant expression
Now substituting these values back into our expression for \(D\):

\[ D = 1 \cdot (-31) - 4 \cdot (-44) + 9 \cdot (-17) \]

Calculating each term:

\[ D = -31 + 176 - 153 \]

### Step 4: Simplify the expression
Now, we simplify:

\[ D = -31 + 176 - 153 = -31 + 23 = -8 \]

Thus, the value of the determinant is

\[ \boxed{-8} \]