For matrix A, what is the characteristic polynomial and the eigenvector values.

To find the characteristic polynomial and eigenvalues of the matrix \(A\), we need to follow these steps:

1. **Compute the characteristic polynomial**: The characteristic polynomial of a matrix \(A\) is given by \(\det(A - \lambda I)\), where \(\lambda\) is a scalar and \(I\) is the identity matrix.

2. **Solve for the eigenvalues**: The eigenvalues are the roots of the characteristic polynomial.

Given the matrix \(A = \begin{bmatrix} 2 & -7 \\ -6 & 3 \end{bmatrix}\):

### Step 1: Compute the Characteristic Polynomial
The characteristic polynomial is found by calculating the determinant of \(A - \lambda I\):

\[ A - \lambda I = \begin{bmatrix} 2 - \lambda & -7 \\ -6 & 3 - \lambda \end{bmatrix} \]

The determinant of this matrix is:

\[
\det(A - \lambda I) = (2 - \lambda)(3 - \lambda) - (-7)(-6)
\]

\[
= (2 - \lambda)(3 - \lambda) - 42
\]

\[
= 6 - 5\lambda + \lambda^2 - 42
\]

\[
= \lambda^2 - 5\lambda - 36
\]

So, the characteristic polynomial is:

\[ \lambda^2 - 5\lambda - 36 \]

### Step 2: Solve for the Eigenvalues
To find the eigenvalues, we solve the characteristic polynomial:

\[ \lambda^2 - 5\lambda - 36 = 0 \]

Using the quadratic formula \(\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -5\), and \(c = -36\):

\[
\lambda = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-36)}}{2(1)}
\]

\[
= \frac{5 \pm \sqrt{25 + 144}}{2}
\]

\[
= \frac{5 \pm \sqrt{169}}{2}
\]

\[
= \frac{5 \pm 13}{2}
\]

Thus, the eigenvalues are:

\[
\lambda_1 = \frac{5 + 13}{2} = 9
\]
\[
\lambda_2 = \frac{5 - 13}{2} = -4
\]

### Eigenvalues
The eigenvalues of the matrix \(A\) are \(\lambda_1 = 9\) and \(\lambda_2 = -4\).

If you need any further details or have questions, feel free to ask!

Here are 8 related questions to explore further:

1. How do you find the eigenvectors corresponding to each eigenvalue?
2. What is the geometric interpretation of eigenvalues and eigenvectors?
3. How can you verify if a given vector is an eigenvector of a matrix?
4. What are the properties of diagonalizable matrices?
5. How do you compute the determinant of a matrix?
6. How does the trace of a matrix relate to its eigenvalues?
7. What is the significance of the characteristic polynomial in linear algebra?
8. How do eigenvalues affect the stability of a system of differential equations?

**Tip:** To find the eigenvectors, substitute each eigenvalue back into the equation \((A - \lambda I)v = 0\) and solve for the vector \(v\).

Discover how to find the characteristic polynomial and eigenvalues of matrix A step-by-step. This problem involves matrix algebra, determinants, and the quadratic formula. Suitable for advanced college-level mathematics.

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