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The problem shown involves diagonalizing the matrix \( A \) and then finding the matrix \( A^4 \). The matrix \( A \) is given as:

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

**Step 1: Finding the Eigenvalues of A**
We start by finding the eigenvalues, \(\lambda\), which satisfy the characteristic equation \(\det(A - \lambda I) = 0\).

\[ A - \lambda I = \begin{bmatrix} 6 - \lambda & -2 \\ 2 & 3 - \lambda \end{bmatrix} \]
\[ \text{Determinant} = (6-\lambda)(3-\lambda) - (-2)(2) = \lambda^2 - 9\lambda + 22 \]

Solving \(\lambda^2 - 9\lambda + 22 = 0\) will give us the eigenvalues.

**Step 2: Finding the Eigenvectors of A**
For each eigenvalue \(\lambda\), solve \((A - \lambda I)\vec{v} = 0\) for the eigenvector \(\vec{v}\).

**Step 3: Diagonalizing A**
Form the matrix \( P \) from the eigenvectors of \( A \) and the diagonal matrix \( D \) from the eigenvalues.

**Step 4: Calculating \( A^4 \)**
\[ A^4 = PDP^{-1}PDP^{-1}PDP^{-1}PDP^{-1} = PD^4P^{-1} \]
Here, \( D^4 \) is simply the diagonal matrix with each eigenvalue raised to the fourth power.

Let's proceed with these calculations now, starting with the eigenvalues.

Diagonalize A = [[6, -2], [-2, 3], [2, -1], [2, 3]] and find A^4. A = 2, 2, 8 with eigenvectors [[2, -1, 1], [0, -1, -1]].

Learn how to diagonalize a matrix A and compute A^4 using eigenvalues and eigenvectors. This detailed example includes key linear algebra concepts such as matrix diagonalization and the power of a matrix, suitable for high school and college students.

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