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To determine which of these matrices is an upper triangular matrix, we first need to recall the definition of an upper triangular matrix. An upper triangular matrix is a square matrix where all elements below the main diagonal are zero. The main diagonal consists of elements from the top-left to the bottom-right.

Let's evaluate each matrix:

1. First matrix:
\[
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-2 & 0 & 0 & 0 \\
-4 & 25 & 0 & 0
\end{bmatrix}
\]
This matrix has non-zero elements below the main diagonal (specifically, -2 and -4), so **this is not** an upper triangular matrix.

2. Second matrix:
\[
\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 7 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\]
This matrix has non-zero elements below the main diagonal (specifically, 1), so **this is not** an upper triangular matrix.

3. Third matrix:
\[
\begin{bmatrix}
2 & -2 & 3 & 5 \\
0 & 5 & 4 & -8 \\
0 & 0 & 1 & 10 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]
All elements below the main diagonal are zero, so **this is an upper triangular matrix**.

4. Fourth matrix:
\[
\begin{bmatrix}
1 & -3 & 21 & 0 \\
0 & 3 & 5 & 11 \\
0 & 0 & -1 & 0
\end{bmatrix}
\]
This is a 3x4 matrix, but not all elements below the main diagonal are zero (specifically, in the 3rd column of the second row), so **this is not** an upper triangular matrix.

### Answer:
The **third matrix** is an upper triangular matrix.

Do you have any questions or would you like further details?

Here are 5 related questions to deepen your understanding:
1. What is the difference between an upper and a lower triangular matrix?
2. How can we identify a diagonal matrix, and how does it relate to upper triangular matrices?
3. In what types of problems are upper triangular matrices useful?
4. What operations can be simplified when a matrix is upper triangular?
5. Can an upper triangular matrix have non-zero elements on its main diagonal?

**Tip:** For any square matrix, if all elements below the main diagonal are zero, it's classified as an upper triangular matrix.

Which of the following is an upper triangular matrix?

Learn how to identify an upper triangular matrix from a set of examples. An upper triangular matrix is a square matrix where all elements below the main diagonal are zero. Suitable for students in Grades 10-12 studying linear algebra and matrix theory.

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