1. Let A=[(4,2),(1,3)] frame the characteristic equation for the given matrix.

To find the characteristic equation of a given matrix \( A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} \), follow these steps:

### 1. Define the Matrix
Given matrix \( A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} \).

### 2. Write the Characteristic Equation
The characteristic equation is found by solving the determinant of \( A - \lambda I \), where \( \lambda \) is an eigenvalue and \( I \) is the identity matrix.

\[
\text{Characteristic Equation:} \quad \text{det}(A - \lambda I) = 0
\]

### 3. Subtract \( \lambda \) from the Diagonal Elements of \( A \)
Subtract \( \lambda \) from each diagonal element of matrix \( A \):

\[
A - \lambda I = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 4-\lambda & 2 \\ 1 & 3-\lambda \end{pmatrix}
\]

### 4. Find the Determinant of \( A - \lambda I \)
Now, compute the determinant of the resulting matrix:

\[
\text{det}(A - \lambda I) = \begin{vmatrix} 4-\lambda & 2 \\ 1 & 3-\lambda \end{vmatrix} = (4-\lambda)(3-\lambda) - (2)(1)
\]

### 5. Simplify the Determinant
Expand and simplify the determinant:

\[
(4-\lambda)(3-\lambda) - 2 = (12 - 4\lambda - 3\lambda + \lambda^2) - 2 = \lambda^2 - 7\lambda + 10
\]

### 6. Write the Characteristic Equation
The characteristic equation is:

\[
\lambda^2 - 7\lambda + 10 = 0
\]

This is the characteristic equation for the matrix \( A \).

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Would you like more details or have any questions about this process? 

Here are some related questions you might find interesting:
1. How do you find the eigenvalues of the matrix \( A \)?
2. What is the significance of the characteristic equation in linear algebra?
3. How do you compute the eigenvectors once you have the eigenvalues?
4. Can you explain the relationship between the trace of the matrix and the characteristic equation?
5. How would the characteristic equation change if the matrix were a \( 3 \times 3 \) matrix?

**Tip:** The roots of the characteristic equation are the eigenvalues of the matrix.

Learn how to frame the characteristic equation for the matrix A = [(4,2),(1,3)] step-by-step. This problem covers linear algebra concepts, including matrices, eigenvalues, and the characteristic equation. Suitable for advanced high school and college students.

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