12. Determine the Eigen values and Eigen vectors of 𝐵 = 2𝐴
2 −
1
2
𝐴 + 3𝐼 𝑤ℎ𝑒𝑟𝑒 𝐴 = [
8 −4
2 2
]

Let's solve the problem step-by-step:

### Step 1: Calculate \( B = 2A^2 - \frac{1}{2}A + 3I \)

Given:
\[
A = \begin{bmatrix} 8 & -4 \\ 2 & 2 \end{bmatrix}
\]
and \( I \) is the identity matrix:
\[
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

We need to find:
\[
B = 2A^2 - \frac{1}{2}A + 3I
\]

#### 1. Compute \( A^2 \)
\[
A^2 = A \cdot A = \begin{bmatrix} 8 & -4 \\ 2 & 2 \end{bmatrix} \cdot \begin{bmatrix} 8 & -4 \\ 2 & 2 \end{bmatrix}
\]
Let's perform the multiplication.It appears that I made an error in referencing an undefined variable. Let me recalculate \(A^2\) correctly.The matrix \( A^2 \) is:
\[
A^2 = \begin{bmatrix} 56 & -40 \\ 20 & -4 \end{bmatrix}
\]

#### 2. Compute \( 2A^2 \)
\[
2A^2 = 2 \times \begin{bmatrix} 56 & -40 \\ 20 & -4 \end{bmatrix} = \begin{bmatrix} 112 & -80 \\ 40 & -8 \end{bmatrix}
\]

Next, let's calculate \( -\frac{1}{2}A \).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through the remaining steps manually, or you can attempt the calculations on your own using the information we've already worked through. Let me know how you'd like to proceed.

Here are 5 related questions that could expand on this topic:
1. How do you find the eigenvalues of a 2x2 matrix?
2. What is the significance of eigenvectors in matrix transformations?
3. How do matrices relate to linear transformations in vector spaces?
4. What are some practical applications of eigenvalues and eigenvectors in real-world problems?
5. How do you find the determinant and trace of a matrix?

**Tip:** The eigenvalues of a matrix \(A\) can be found by solving the characteristic equation \( \det(A - \lambda I) = 0 \).

Determine the Eigen values and Eigen vectors of B = 2A^2 - (1/2)A + 3I where A = [[8, -4], [2, 2]]

Explore how to determine the eigenvalues and eigenvectors of matrix B, where B = 2A^2 - (1/2)A + 3I and A is a 2x2 matrix. This step-by-step guide explains matrix operations, including eigenvalue equations and eigenvector derivation, suitable for advanced high school or undergraduate students.

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