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The problem asks us to determine matrix \( A \), given the following information:

1. Matrix \( [T^A_B] \) is given as:

\[
[T^A_B] = \begin{pmatrix}
-1 & 4 \\
4 & -11
\end{pmatrix}
\]

2. The set \( B \) is given as:

\[
B = \{(3, 5), (1, 2)\}
\]

To solve for \( A \), we need to understand that the matrix \( [T^A_B] \) represents the transformation matrix from basis \( B \) to the basis \( A \). Specifically, \( [T^A_B] \) expresses how the vectors in basis \( B \) are transformed into basis \( A \).

We can represent the matrix \( A \) using the transformation matrix as follows:

\[
A = [T^A_B] \times B
\]

Let's first express the matrix \( B \) using the given vectors:

\[
B = \begin{pmatrix}
3 & 1 \\
5 & 2
\end{pmatrix}
\]

We will multiply matrix \( [T^A_B] \) with the matrix representing \( B \) to determine matrix \( A \).

Let me calculate that.The matrix \( A \) is determined as:

\[
A = \begin{pmatrix}
17 & 7 \\
-43 & -18
\end{pmatrix}
\]

Would you like further details on the process, or do you have any questions?

Here are five related questions for further exploration:

1. What is the geometric interpretation of the transformation matrix \( [T^A_B] \)?
2. How do we find the inverse of a transformation matrix?
3. What happens if the matrix \( [T^A_B] \) is not invertible?
4. How can we verify if the vectors in set \( B \) form a valid basis?
5. What is the relationship between different bases and the transformation matrix?

**Tip:** When dealing with transformations between different bases, understanding the basis vectors is key to interpreting the transformation matrix accurately.

Sabendo que [T^A_B = [-1 4; 4 -11]] e B = {(3,5),(1,2)}, determine A.

Learn how to determine matrix A using a transformation matrix [T^A_B] and a set basis B. This problem explores key linear algebra concepts, including matrix multiplication and basis change. Suitable for undergraduate students studying linear algebra.

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