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To find the transpose \( A^T \) of the given matrix \( A \), we switch the rows and columns of \( A \). Here's the original matrix:

\[
A = \begin{bmatrix}
1 & -8 & 4 \\
2 & -9 & 5 \\
0 & -2 & 1
\end{bmatrix}
\]

The transpose \( A^T \) is:

\[
A^T = \begin{bmatrix}
1 & 2 & 0 \\
-8 & -9 & -2 \\
4 & 5 & 1
\end{bmatrix}
\]

So the result is:

\[
A^T = \begin{bmatrix}
1 & 2 & 0 \\
-8 & -9 & -2 \\
4 & 5 & 1
\end{bmatrix}
\]

Do you have any questions or would you like further details on how matrix transposition works?

Here are 5 related questions:
1. How do you compute the transpose of a non-square matrix?
2. What properties remain the same after a matrix is transposed?
3. How does transposing affect the determinant of a matrix?
4. What is the transpose of the product of two matrices?
5. Can the transpose of a matrix change the rank of a matrix?

**Tip:** When you transpose a matrix, any diagonal elements remain unchanged because they are part of both row and column for that index.

Given A = [[1, -8, 4], [2, -9, 5], [0, -2, 1]], find A^T.

Learn how to transpose the matrix A = [[1, -8, 4], [2, -9, 5], [0, -2, 1]] by switching its rows and columns. This step-by-step solution demonstrates key linear algebra concepts for students in Grades 10-12.

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