We are asked to find the determinant of matrix $A$ by expanding along the first column. The matrix $A$ is given as:

$A = \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix}$

To compute the determinant of this 3x3 matrix by expanding along the first column, we use the cofactor expansion method.

The determinant $\text{det}(A)$ is:

$\text{det}(A) = 1 \cdot \begin{vmatrix} 3 & 2 \\ 10 & 9 \end{vmatrix} - 1 \cdot \begin{vmatrix} 3 & 4 \\ 10 & 9 \end{vmatrix} + 4 \cdot \begin{vmatrix} 3 & 4 \\ 3 & 2 \end{vmatrix}$

Step-by-step calculation:

1. First 2x2 matrix:

$\begin{vmatrix} 3 & 2 \\ 10 & 9 \end{vmatrix} = (3 \times 9) - (2 \times 10) = 27 - 20 = 7$

2. Second 2x2 matrix:

$\begin{vmatrix} 3 & 4 \\ 10 & 9 \end{vmatrix} = (3 \times 9) - (4 \times 10) = 27 - 40 = -13$

3. Third 2x2 matrix:

$\begin{vmatrix} 3 & 4 \\ 3 & 2 \end{vmatrix} = (3 \times 2) - (4 \times 3) = 6 - 12 = -6$

Substituting the values back into the expansion formula:

$\text{det}(A) = 1 \cdot 7 - 1 \cdot (-13) + 4 \cdot (-6)$

$\text{det}(A) = 7 + 13 - 24 = -4$

Thus, the determinant of matrix $A$ is $\boxed{-4}$.

Would you like a more detailed explanation of any step? Here are some related questions to explore:

1. How do you calculate the determinant of larger matrices (4x4 and beyond)?
2. What is the cofactor expansion method for other rows or columns?
3. How does the determinant help in solving systems of linear equations?
4. What happens to the determinant if two rows or columns are swapped?
5. How can the determinant of a matrix indicate if it is invertible?

Tip: The determinant of a matrix helps in identifying if a matrix is invertible. If the determinant is 0, the matrix is singular and does not have an inverse.