The eigenvalues for
𝐴=⎛⎝⎜⎜41015−10−14⎞⎠⎟⎟
are 4,3
, and 6
.The eigenvalues have respective eigenvectors ⎛⎝⎜⎜101⎞⎠⎟⎟,⎛⎝⎜⎜1−1−1⎞⎠⎟⎟,
 and ⎛⎝⎜⎜12−1⎞⎠⎟⎟
 (or their multiples). explain how we know that those are their respective eigenvectors

To explain how we know that the given eigenvectors correspond to the given eigenvalues of matrix \( A \), let's go step by step. The key concept here is the **eigenvalue equation**:

\[
A \mathbf{v} = \lambda \mathbf{v}
\]

where:
- \( A \) is the matrix,
- \( \mathbf{v} \) is the eigenvector,
- \( \lambda \) is the eigenvalue.

In simpler terms, multiplying the matrix \( A \) by an eigenvector \( \mathbf{v} \) results in a scalar multiple of that vector, where the scalar is the eigenvalue \( \lambda \).

The matrix \( A \) is given as:

\[
A = \begin{pmatrix} 
4 & 10 & 1 \\
0 & 15 & -1 \\
0 & -1 & 4
\end{pmatrix}
\]

The eigenvalues of this matrix are \( 4 \), \( 3 \), and \( 6 \), and the corresponding eigenvectors are provided. To confirm that each vector is indeed an eigenvector for its respective eigenvalue, we check if the matrix \( A \) times each eigenvector gives the correct scalar multiple of the vector.

### 1. Eigenvalue \( \lambda = 4 \) and eigenvector \( \mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} \)

We compute:

\[
A \mathbf{v}_1 = A \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 4 & 10 & 1 \\ 0 & 15 & -1 \\ 0 & -1 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}
\]

Carrying out the multiplication:

\[
A \mathbf{v}_1 = \begin{pmatrix} 4(1) + 10(0) + 1(1) \\ 0(1) + 15(0) - 1(1) \\ 0(1) - 1(0) + 4(1) \end{pmatrix} = \begin{pmatrix} 5 \\ -1 \\ 4 \end{pmatrix}
\]

Since we are checking for the eigenvalue \( \lambda = 4 \), we need to verify if:

\[
A \mathbf{v}_1 = 4 \mathbf{v}_1 = 4 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 4 \\ 0 \\ 4 \end{pmatrix}
\]

Clearly, \( \begin{pmatrix} 5 \\ -1 \\ 4 \end{pmatrix} \neq \begin{pmatrix} 4 \\ 0 \\ 4 \end{pmatrix} \), so there seems to be an error in this specific case.

Would you like me to proceed with recalculating the other cases, or does this mismatch suggest a need for review of your problem setup? You may also upload any visual reference for clarity.

Here are 5 relative questions:
1. What is the method for finding eigenvalues of a matrix?
2. How do we solve for eigenvectors given eigenvalues?
3. What does it mean if a vector is not an eigenvector of a matrix?
4. How does matrix multiplication work in verifying eigenvalue equations?
5. Can you explain the geometric interpretation of eigenvectors?

Tip: Always verify matrix computations step by step to avoid errors when dealing with eigenvalue and eigenvector problems.

The eigenvalues for A = ⎛⎝⎜⎜41015−10−14⎞⎠⎟⎟ are 4, 3, and 6. The eigenvalues have respective eigenvectors ⎛⎝⎜⎜101⎞⎠⎟⎟, ⎛⎝⎜⎜1−1−1⎞⎠⎟⎟, and ⎛⎝⎜⎜12−1⎞⎠⎟⎟ (or their multiples). Explain how we know that those are their respective eigenvectors.

Learn how to verify the eigenvectors corresponding to the eigenvalues of a 3x3 matrix A with eigenvalues 4, 3, and 6. This solution walks through the key steps of matrix multiplication and the eigenvalue equation to explain how each eigenvector corresponds to its respective eigenvalue.

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