Question

Math

Posted 5 months ago

```
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.
```

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Answer from Sia

Posted 5 months ago

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.

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