Q.3 For the below matrices
i) Find the eigen values and the eigen vectors.
ii) Verify Cayley-Hamilton Theorem.
iii)Check if they are diagonalisable, if yes find the diagonal and transforming matrix.
                       A= [■(9&-1&9@-1&-1&3@-7&1&-7)]


### Matrix Given:
\[ A = \begin{bmatrix} 9 & -1 & 9 \\ -1 & -1 & 3 \\ -7 & 1 & -7 \end{bmatrix} \]

Let's address each part of the question:

### Part i) Find the Eigenvalues and Eigenvectors

To find the eigenvalues of matrix \( A \), we solve the characteristic equation:

\[
\text{det}(A - \lambda I) = 0
\]

Where \( \lambda \) is the eigenvalue, \( I \) is the identity matrix, and \( \text{det} \) represents the determinant.

### Part ii) Verify Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. This means that if the characteristic equation of matrix \( A \) is:

\[
p(\lambda) = \text{det}(A - \lambda I) = 0
\]

Then, the matrix \( A \) satisfies:

\[
p(A) = 0
\]

### Part iii) Check if the Matrix is Diagonalizable

A matrix is diagonalizable if there are \( n \) linearly independent eigenvectors for an \( n \times n \) matrix. If \( A \) is diagonalizable, we can write:

\[
A = P D P^{-1}
\]

Where \( D \) is the diagonal matrix of eigenvalues, and \( P \) is the matrix whose columns are the corresponding eigenvectors of \( A \).

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Here are 5 related questions that might interest you:
1. How do you manually solve for eigenvalues and eigenvectors?
2. Can you explain the Cayley-Hamilton theorem in detail?
3. What is the process for checking if a matrix is diagonalizable?
4. What are the properties of matrices that make them diagonalizable?
5. How do you calculate the inverse of a matrix?

**Tip**: Diagonalizing a matrix is a useful method for simplifying matrix powers and solving systems of linear differential equations.

Q.3 For the below matrices i) Find the eigen values and the eigen vectors. ii) Verify Cayley-Hamilton Theorem. iii) Check if they are diagonalisable, if yes find the diagonal and transforming matrix. A = [9, -1, 9; -1, -1, 3; -7, 1, -7]

Explore how to find eigenvalues and eigenvectors for the matrix A = [9, -1, 9; -1, -1, 3; -7, 1, -7], verify the Cayley-Hamilton Theorem, and check if the matrix is diagonalizable. This detailed explanation covers key concepts in linear algebra, suitable for undergraduate-level students.

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