Let's find the determinant of the given matrix step-by-step.
The given matrix is:
[tex]\[
\begin{vmatrix}
1 & 2 & 4 \\
1 & 3 & 9 \\
1 & 4 & 16
\end{vmatrix}
\][/tex]
### Step-by-Step Calculation:
1. Choose a row or a column to expand along.
To simplify our calculation, let's expand along the first row because it contains simple numbers (mostly 1's and a minor 2).
2. Apply the cofactor expansion along the first row:
[tex]\[
\text{Det} = 1 \cdot \begin{vmatrix}
3 & 9 \\
4 & 16
\end{vmatrix}
- 2 \cdot \begin{vmatrix}
1 & 9 \\
1 & 16
\end{vmatrix}
+ 4 \cdot \begin{vmatrix}
1 & 3 \\
1 & 4
\end{vmatrix}
\][/tex]
3. Calculate the determinants of the 2x2 submatrices:
- For [tex]\(\begin{vmatrix}
3 & 9 \\
4 & 16
\end{vmatrix}\)[/tex]:
[tex]\[
(3 \cdot 16) - (9 \cdot 4) = 48 - 36 = 12
\][/tex]
- For [tex]\(\begin{vmatrix}
1 & 9 \\
1 & 16
\end{vmatrix}\)[/tex]:
[tex]\[
(1 \cdot 16) - (9 \cdot 1) = 16 - 9 = 7
\][/tex]
- For [tex]\(\begin{vmatrix}
1 & 3 \\
1 & 4
\end{vmatrix}\)[/tex]:
[tex]\[
(1 \cdot 4) - (3 \cdot 1) = 4 - 3 = 1
\][/tex]
4. Substitute these values back into our cofactor expansion:
[tex]\[
\text{Det} = 1 \cdot 12 - 2 \cdot 7 + 4 \cdot 1
\][/tex]
Simplify:
[tex]\[
\text{Det} = 12 - 14 + 4
\][/tex]
[tex]\[
\text{Det} = 12 - 14 + 4 = 2
\][/tex]
So, the determinant of the matrix is [tex]\(2\)[/tex]. Therefore, the correct option is:
[tex]\[
\boxed{2}
\][/tex]
