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

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

### Step 1: Find the matrix \( A - \lambda I \)
Here, \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix. For a 2x2 matrix, the identity matrix \( I \) is:

\[
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

So, \( A - \lambda I \) is calculated as:

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

### Step 2: Find the determinant of \( A - \lambda I \)
The characteristic equation is found by setting the determinant of \( A - \lambda I \) equal to zero.

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

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as:

\[
\text{det} = ad - bc
\]

So, for our matrix:

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

### Step 3: Simplify the equation
Now, expand and simplify:

\[
\text{det}(A - \lambda I) = (4 - \lambda)(3 - \lambda) - 2 = (12 - 4\lambda - 3\lambda + \lambda^2) - 2 = \lambda^2 - 7\lambda + 12 - 2
\]

\[
\text{det}(A - \lambda I) = \lambda^2 - 7\lambda + 10
\]

### Step 4: Form the characteristic equation
The characteristic equation is:

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

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

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you find the eigenvalues using the characteristic equation?
2. What are the steps to find eigenvectors once the eigenvalues are known?
3. How does the characteristic equation relate to the trace and determinant of the matrix?
4. Can the characteristic equation be used for larger matrices, such as 3x3?
5. What is the physical significance of eigenvalues and eigenvectors in real-world applications?

**Tip:** When solving the characteristic equation, double-check the signs and coefficients to avoid mistakes.

Learn how to find the characteristic equation of matrix A = [[4, 2], [1, 3]] step-by-step. This problem covers matrix algebra concepts, including eigenvalues and determinants of a 2x2 matrix. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation