https://file.aiquickdraw.com/mathbot/file/17288829878405sjot534.jpg

The matrix provided is:

\[
\begin{pmatrix}
1 & 0 & -5 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

This matrix is a **3x3 elementary matrix**. We are tasked with finding its determinant.

### Key Property:
- The determinant of an upper triangular matrix (where all elements below the diagonal are zero) is the product of the diagonal elements.

In this matrix, the diagonal elements are 1, 1, and 1.

\[
\text{determinant} = 1 \times 1 \times 1 = 1
\]

Thus, the determinant of the matrix is **1**.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What are elementary matrices, and how do they affect determinants?
2. How do matrix transformations like row operations affect the determinant?
3. Why is the determinant of an upper triangular matrix equal to the product of its diagonal entries?
4. How does the determinant help in understanding invertibility of matrices?
5. What are the properties of matrices with determinants equal to zero?

**Tip:** The determinant of a matrix tells whether the matrix is invertible (non-zero determinant) or singular (zero determinant).

What is the determinant of the 3x3 elementary matrix below? (1 0 -5; 0 1 0; 0 0 1)

Learn how to find the determinant of a 3x3 elementary matrix in upper triangular form. Explore matrix properties, determinants, and related theorems, suitable for linear algebra courses at the undergraduate level.

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