Let's solve the problem based on the given matrices.

1. Determinant of $A$

Matrix $A$ is given as: $A = \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix}$

To find the determinant of $A$ by expanding along the first column, we use the cofactor expansion formula. For a 3x3 matrix: $\text{det}(A) = a_{11} \cdot \text{det}(A_{11}) - a_{21} \cdot \text{det}(A_{21}) + a_{31} \cdot \text{det}(A_{31})$ where $A_{ij}$ is the minor matrix obtained by removing the i-th row and j-th column.

Now, let's expand and compute this.

2. $(A^2 - B^T)C$

Matrix $B$ and vector $C$ are given as: $B = \begin{pmatrix} 7 & 13 & -6 \\ -1 & -7 & 2 \\ -2 & 2 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$

• First, find $A^2$ (the square of matrix $A$).
• Then, compute the transpose of matrix $B$, denoted $B^T$.
• Subtract $B^T$ from $A^2$.
• Finally, multiply the resulting matrix by vector $C$.

I will now perform the calculations.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you step-by-step on how to calculate both the determinant and the matrix operations manually. Let me know if you'd prefer that or have any other questions!

Here are five related questions you might find interesting:

1. How can you verify the determinant of a 3x3 matrix using another expansion method?
2. What are the properties of matrix transposition?
3. How do you interpret the multiplication of a matrix with a vector in applications?
4. What is the geometrical interpretation of the determinant of a matrix?
5. How do matrix powers (like $A^2$) relate to transformations?

Tip: When expanding a determinant along a row or column, focus on the one with the most zeros to simplify calculations.