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The image you provided involves finding a vector \( \mathbf{x} \) such that when transformed by a matrix \( A \), the result equals a given vector. Specifically, it asks to find the vector \( \mathbf{x} \) whose image under the transformation \( T(\mathbf{x}) = A\mathbf{x} \) results in a given vector.

### Problem Breakdown:
- The matrix \( A \) is given as:
\[
A = \begin{pmatrix}
6 & -2 & -3 \\
5 & -4 & 2 \\
-42 & 11 & 6
\end{pmatrix}
\]
- The transformation is described by \( T(\mathbf{x}) = A\mathbf{x} \), where the result is:
\[
\mathbf{b} = \begin{pmatrix} 0 \\ 5 \\ 39 \end{pmatrix}
\]
- We need to solve for the vector \( \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \) such that:
\[
A\mathbf{x} = \mathbf{b}
\]

### Solution Steps:
1. Set up the system of equations based on \( A\mathbf{x} = \mathbf{b} \).
2. Solve the system of equations to find the vector \( \mathbf{x} \).

Let me calculate that for you.The solution to the system \( A\mathbf{x} = \mathbf{b} \) is the vector:

\[
\mathbf{x} = \begin{pmatrix} -2.505 \\ -5.165 \\ -1.567 \end{pmatrix}
\]

Thus, the vector whose image under the transformation is \( \mathbf{b} = \begin{pmatrix} 0 \\ 5 \\ 39 \end{pmatrix} \) is approximately:

\[
\mathbf{x} = \begin{pmatrix} -2.505 \\ -5.165 \\ -1.567 \end{pmatrix}
\]

Additionally, based on the system of equations and the matrix, this solution is unique, as the matrix \( A \) is invertible (its determinant is non-zero).

Would you like any additional details or explanations on the steps?

Here are five related questions you might find helpful:
1. How can we determine if a matrix is invertible?
2. What does it mean for a transformation to be one-to-one or onto?
3. How do you compute the determinant of a 3x3 matrix?
4. Can we always solve \( A\mathbf{x} = \mathbf{b} \) for any matrix \( A \)?
5. How can eigenvalues be related to solving systems of linear equations?

**Tip:** If a system of linear equations has a unique solution, the matrix of coefficients must be invertible, meaning its determinant is non-zero.

Find a vector x such that T(x) = Ax, where the matrix A and the vector b are given, and solve for x.

Learn how to solve the linear equation Ax = b using matrix multiplication. This problem covers key linear algebra concepts, including vector spaces and the Invertible Matrix Theorem. Suitable for high school students in Grades 11-12.

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