https://file.aiquickdraw.com/mathbot/file/1729448202655w3h3oj1n.jpg

The image shows a matrix and a question regarding the development of its determinant by expanding along the first column. The matrix given is:

\[
\begin{pmatrix}
1 & 2 & -3 & 4 \\
0 & 3 & 1 & 5 \\
-1 & 0 & 7 & -2 \\
0 & 1 & 3 & 5
\end{pmatrix}
\]

The question asks to expand the determinant using the first column and select the correct expression.

### Step-by-Step Process:
When expanding the determinant along the first column (Laplace expansion), we take the element from the first column, multiply it by the cofactor, and do this for every row in that column.

1. **Element at (1,1):** The first element is \(1\). We remove the first row and first column, giving the cofactor matrix:
   \[
   \begin{pmatrix}
   3 & 1 & 5 \\
   0 & 7 & -2 \\
   1 & 3 & 5
   \end{pmatrix}
   \]
   The contribution from this element is \( 1 \times \text{det of the cofactor matrix} \).

2. **Element at (2,1):** The second element is \(0\), so the contribution is \(0\).

3. **Element at (3,1):** The third element is \(-1\). Remove the third row and the first column, giving the cofactor matrix:
   \[
   \begin{pmatrix}
   2 & -3 & 4 \\
   3 & 1 & 5 \\
   1 & 3 & 5
   \end{pmatrix}
   \]
   The contribution from this element is \( (-1) \times \text{det of the cofactor matrix} \).

4. **Element at (4,1):** The fourth element is \(0\), so again the contribution is \(0\).

Thus, the expression for the determinant is:

\[
1 \cdot \text{det}\left(\begin{pmatrix} 3 & 1 & 5 \\ 0 & 7 & -2 \\ 1 & 3 & 5 \end{pmatrix}\right) + (-1) \cdot \text{det}\left(\begin{pmatrix} 2 & -3 & 4 \\ 3 & 1 & 5 \\ 1 & 3 & 5 \end{pmatrix}\right)
\]

### Correct Option:
This matches with option **(b)** from the image, which shows:
\[
= 1 \cdot \text{det}\begin{pmatrix} 3 & 1 & 5 \\ 0 & 7 & -2 \\ 1 & 3 & 5 \end{pmatrix} + (-1) \cdot \text{det}\begin{pmatrix} 2 & -3 & 4 \\ 3 & 1 & 5 \\ 1 & 3 & 5 \end{pmatrix}
\]

Let me know if you'd like further details or explanation!

Here are some follow-up questions:
1. What is the Laplace expansion of a 4x4 matrix in general?
2. How can you compute the determinant of a 3x3 matrix using cofactor expansion?
3. What is the meaning of a cofactor in determinant calculations?
4. What are the properties of determinants of square matrices?
5. How does changing a column in a matrix affect its determinant?

**Tip:** When calculating determinants using cofactor expansion, always check for rows or columns with zeroes. Expanding along them simplifies the calculations!

Si desenvolupem el determinant de la següent matriu per la primera columna, quina expressió és la correcta?

\[
\begin{pmatrix}
1 & 2 & -3 & 4 \\
0 & 3 & 1 & 5 \\
-1 & 0 & 7 & -2 \\
0 & 1 & 3 & 5
\end{pmatrix}
\]

Learn how to develop a 4x4 matrix determinant along the first column using cofactor expansion. The problem uses Laplace expansion with detailed steps and shows how to calculate the determinant of 3x3 cofactor matrices.

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