https://file.aiquickdraw.com/mathbot/file/1730025042617st6awbbi.jpg

To find the determinant of the \( 3 \times 3 \) matrix \( A \) shown in the image, we can use the cofactor expansion along the first row:

\[
A = \begin{vmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{vmatrix}
\]

For a \( 3 \times 3 \) matrix \( A \), the determinant is calculated as:

\[
|A| = a_{11} \cdot \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12} \cdot \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13} \cdot \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}
\]

For this specific matrix:

- \( a_{11} = 1 \), \( a_{12} = 2 \), \( a_{13} = 3 \)
- Sub-matrices:

1. **First Term**:
   \[
   a_{11} \cdot \begin{vmatrix} 2 & 4 \\ 0 & 5 \end{vmatrix} = 1 \cdot (2 \cdot 5 - 4 \cdot 0) = 1 \cdot 10 = 10
   \]

2. **Second Term**:
   \[
   - a_{12} \cdot \begin{vmatrix} 0 & 4 \\ 0 & 5 \end{vmatrix} = -2 \cdot (0 \cdot 5 - 4 \cdot 0) = -2 \cdot 0 = 0
   \]

3. **Third Term**:
   \[
   + a_{13} \cdot \begin{vmatrix} 0 & 2 \\ 0 & 0 \end{vmatrix} = 3 \cdot (0 \cdot 0 - 2 \cdot 0) = 3 \cdot 0 = 0
   \]

Adding these up:

\[
|A| = 10 + 0 + 0 = 10
\]

So, the determinant of matrix \( A \) is \( |A| = 10 \).

Would you like more details or have any questions?

---

1. How do you calculate determinants for larger matrices (e.g., \( 4 \times 4 \))?
2. What is cofactor expansion in the context of matrix determinants?
3. Can you explain why terms with zero rows or columns contribute zero to the determinant?
4. How would the determinant calculation change if one of the diagonal elements were zero?
5. How are determinants applied in solving systems of linear equations?

**Tip:** When calculating determinants, always look for rows or columns with zeros, as they can simplify the process.

Calculate the determinant of the matrix A given by \( A = \begin{vmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{vmatrix} \).

Learn how to calculate the determinant of the 3x3 matrix \( A = \begin{vmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{vmatrix} \) with a step-by-step cofactor expansion. This guide covers key concepts in matrix algebra and cofactor expansion, suitable for Grades 11-12.

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