What is the characteristic equation of the matrix A = (-5 5 / 4 8)

To find the **characteristic equation** of a matrix, we use the formula:

\[
\text{det}(A - \lambda I) = 0
\]

Where \( A \) is the given matrix, \( \lambda \) represents the eigenvalues, and \( I \) is the identity matrix of the same dimension as \( A \).

Given matrix \( A = \begin{pmatrix} -5 & 5 \\ 4 & 8 \end{pmatrix} \), we now calculate the characteristic equation.

### Step 1: Subtract \( \lambda I \) from \( A \)

\[
A - \lambda I = \begin{pmatrix} -5 & 5 \\ 4 & 8 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -5 - \lambda & 5 \\ 4 & 8 - \lambda \end{pmatrix}
\]

### Step 2: Calculate the determinant

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[
\text{det}(A) = ad - bc
\]

For the matrix \( A - \lambda I = \begin{pmatrix} -5 - \lambda & 5 \\ 4 & 8 - \lambda \end{pmatrix} \), the determinant is:

\[
\text{det}(A - \lambda I) = (-5 - \lambda)(8 - \lambda) - (5)(4)
\]

Expanding this expression:

\[
= (-5 - \lambda)(8 - \lambda) - 20
\]
\[
= (-40 + 5\lambda - 8\lambda - \lambda^2) - 20
\]
\[
= -\lambda^2 - 3\lambda - 60
\]

### Step 3: Set the determinant equal to 0

\[
-\lambda^2 - 3\lambda - 60 = 0
\]

Multiplying through by \(-1\) to simplify:

\[
\lambda^2 + 3\lambda + 60 = 0
\]

Thus, the **characteristic equation** of matrix \( A \) is:

\[
\lambda^2 + 3\lambda + 60 = 0
\]

Would you like further details on solving this equation or its interpretation? 

Here are some related questions:
1. What are the eigenvalues of matrix \( A \)?
2. How does the discriminant of this quadratic affect the nature of the eigenvalues?
3. How do you compute the eigenvectors from the eigenvalues?
4. What is the trace and determinant of the matrix and how are they related to the characteristic equation?
5. How would this process change for a \( 3 \times 3 \) matrix?

**Tip:** For a 2x2 matrix, the sum of the eigenvalues equals the trace, and their product equals the determinant of the matrix.

What is the characteristic equation of the matrix A = (-5 5 / 4 8)?

Learn how to find the characteristic equation of the matrix A = (-5 5 / 4 8) using eigenvalue theory and matrix algebra. This detailed solution covers calculating the determinant of a 2x2 matrix and solving the resulting quadratic equation. Suitable for advanced high school or undergraduate students.

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