Let's break down the questions based on the image provided:

1. Find the determinant of matrix $A$ by expanding the first column.

The matrix $A$ is: $A = \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix}$ We will expand along the first column to compute the determinant.

$\text{det}(A) = 1 \cdot \text{det}\begin{pmatrix} 3 & 2 \\ 10 & 9 \end{pmatrix} - 1 \cdot \text{det}\begin{pmatrix} 3 & 4 \\ 10 & 9 \end{pmatrix} + 4 \cdot \text{det}\begin{pmatrix} 3 & 4 \\ 3 & 2 \end{pmatrix}$ Now, we'll compute the 2x2 determinants.

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

The matrices provided are: $A = \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix}, \quad B = \begin{pmatrix} 7 & 13 & -6 \\ -1 & -7 & 2 \\ -2 & 2 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$

1. Compute $A^2 = A \times A$.
2. Find the transpose of $B$, i.e., $B^T$.
3. Compute $A^2 - B^T$.
4. Finally, multiply the result by $C$.

Let me go ahead and calculate both of these steps.