https://file.aiquickdraw.com/mathbot/file/17260593352588v23ppc7.jpg

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.

Find determinant of A by expanding the first column, and evaluate (A^2 - B^T)C.

Step-by-step solution to finding the determinant of a 3x3 matrix A using column expansion, and evaluating the expression (A^2 - B^T)C. This problem involves matrix operations such as determinant, transpose, and matrix-vector multiplication. Suitable for university-level students studying Linear Algebra.

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