Given a matrix $A$, where
$$
A=\left[\begin{array}{lll}
2 & 2 & 4 \
1 & 3 & 5 \
2 & 3 & 4
\end{array}\right]
$$

Tasks:
1. Find the Eigenvalues of $A$ : Determine the eigenvalues of matrix $A$.
2. Find the Eigenvectors Corresponding to Each Eigenvalue: For each eigenvalue found in Task 1, find the corresponding eigenvectors.
3. Diagonalize $A$, if Possible: If matrix $A$ is diagonalizable, find a matrix $P$ consisting of the eigenvectors of $A$ and a diagonal matrix $D$ such that $A=P D P^{-1}$. If $A$ is not diagonalizable, explain why.

Question

Given a matrix $A$, where
$$
A=\left[\begin{array}{lll}
2 & 2 & 4 \
1 & 3 & 5 \
2 & 3 & 4
\end{array}\right]
$$

Tasks:
1. Find the Eigenvalues of $A$ : Determine the eigenvalues of matrix $A$.
2. Find the Eigenvectors Corresponding to Each Eigenvalue: For each eigenvalue found in Task 1, find the corresponding eigenvectors.
3. Diagonalize $A$, if Possible: If matrix $A$ is diagonalizable, find a matrix $P$ consisting of the eigenvectors of $A$ and a diagonal matrix $D$ such that $A=P D P^{-1}$. If $A$ is not diagonalizable, explain why.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the eigenvalues of matrix $ A $, we calculate the characteristic polynomial and solve for the values of $ \lambda $ that satisfy the equation $ \det(A - \lambda I) = 0 $. ‖ ‖ step 2 ⋮ The eigenvalues calculated from the asksia-ll calculator are $ \lambda_1 \approx 8.80916 $, $ \lambda_2 \approx 0.926209 $, and $ \lambda_3 \approx -0.735373 $. ‖ ‖ step 3 ⋮ To find the eigenvectors corresponding to each eigenvalue, we solve the system $ (A - \lambda_i I) \mathbf{v} = \mathbf{0} $ for each $ \lambda_i $. ‖ ‖ step 4 ⋮ The eigenvectors calculated from the asksia-ll calculator are $ \mathbf{v}_1 \approx (0.885, 1.01305, 1) $, $ \mathbf{v}_2 \approx (7.51633, -6.03548, 1) $, and $ \mathbf{v}_3 \approx (-0.601327, -1.17757, 1) $. ‖ ‖ step 5 ⋮ To diagonalize matrix $ A $, we form matrix $ P $ with the eigenvectors as columns and matrix $ D $ as the diagonal matrix with the corresponding eigenvalues. ‖ ‖ step 6 ⋮ The diagonalization of matrix $ A $ is given by $ A = PDP^{-1} $, where $ P $ and $ D $ are as follows: $ P \approx \begin{pmatrix} -0.601327 & 7.51633 & 0.885 \ -1.17757 & -6.03548 & 1.01305 \ 1 & 1 & 1 \end{pmatrix} $, $ D \approx \begin{pmatrix} -0.735373 & 0 & 0 \ 0 & 0.926209 & 0 \ 0 & 0 & 8.80916 \end{pmatrix} $, and $ P^{-1} \approx \begin{pmatrix} -0.281905 & -0.265219 & 0.518168 \ 0.0876139 & -0.0594455 & -0.0173167 \ 0.194292 & 0.324665 & 0.499149 \end{pmatrix} $. ‖ ∻Answer∻ ⚹ The matrix $ A $ is diagonalizable, and the diagonalization is given by $ A = PDP^{-1} $ with matrices $ P $, $ D $, and $ P^{-1} $ as calculated above. ⚹ ∻Key Concept∻ ⚹Diagonalization of a matrix⚹ ∻Explanation∻ ⚹Diagonalization involves finding a matrix $ P $ of eigenvectors and a diagonal matrix $ D $ of eigenvalues such that $ A = PDP^{-1} $. This process simplifies matrix operations and is possible when matrix $ A $ has a full set of linearly independent eigenvectors.⚹◊What is the formula used to calculate the determinant of a matrix?⍭ Generate me a similar question◊

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