https://file.aiquickdraw.com/mathbot/file/1726109647931svr185j9.jpg

To solve this problem, we need to find the transpose of the given matrix:

\[
\begin{bmatrix}
1 & 0 & -5 \\
2 & 3 & 4 \\
-4 & 7 & 9
\end{bmatrix}
\]

### Step-by-step solution:
The transpose of a matrix is obtained by swapping the rows with the columns.

So, let's transpose the matrix:

- The first row \([1, 0, -5]\) becomes the first column.
- The second row \([2, 3, 4]\) becomes the second column.
- The third row \([-4, 7, 9]\) becomes the third column.

Thus, the transpose of the matrix is:

\[
\begin{bmatrix}
1 & 2 & -4 \\
0 & 3 & 7 \\
-5 & 4 & 9
\end{bmatrix}
\]

Now, let's check which of the given options matches this result.

The second option is:

\[
\begin{bmatrix}
1 & 2 & -4 \\
0 & 3 & 7 \\
-5 & 4 & 9
\end{bmatrix}
\]

This is the correct answer.

### Answer: The second option is correct.

Do you need further details or have any questions?

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Here are five related questions to expand your understanding:
1. How do you compute the transpose of a 2x2 matrix?
2. What are the properties of a matrix transpose?
3. Can the transpose of a matrix change its determinant?
4. What is the transpose of a column vector?
5. How is the transpose used in solving systems of linear equations?

**Tip:** When transposing, simply switch the position of the matrix elements from row to column to avoid confusion.

Which is the transpose of the matrix [ [1, 0, -5], [2, 3, 4], [-4, 7, 9] ]?

Learn how to find the transpose of a 3x3 matrix, including step-by-step instructions for swapping rows and columns. This example is suitable for students in Grades 9-12 and covers core concepts of linear algebra.

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