Question

2. Find all eigenvalues and corresponding linearly independent eigenvectors of A = [2 0 3 4] (Its a 2x2 matrix)

4. Find all eigenvalues and corresponding linearly independent eigenvectors of A = [1 0 1 0 2 3 0 0 3] (Its's a 3x3 matrix)

6. Find all eigenvalues and corresponding eigenvectors of A =    1 2 3 0 1 2 0 0 1    .(Its a 3x3 matrix)

Answer 1

We have to find here the eigenvalues and their corresponding eigenvectors;

__Part 1)__

Given matrix is;

Eigenvalues can be find by subtracting "" from from the diagonal of the given matrix.

Now finding the determinant of the matrix we get;

Hence;

Now we will find the eigenvectors of the obtained eigenvalues;

When  

We get;

Taking [] we get;

Therefore when eigenvalue is 2

eigenvector is;

Now finding eigenvector when

Hence we get;

If we take

then

Therefore eigevectors will be;

Hence;

When eigenvalues is 2;

eigenvector is;

When eigenvalue is 4;

eigenvector is;

__Part 2)__

Given matrix is;

Finding eigenvalue of the given matrix;

Determinant will be;

Now finding eigenvectors  when  

Hence we get;

If we take

We get;

Therefore eigenvectors will be;

Now finding eigenvectors  when  

Hence we get;

If we take then, we get;

Therefore eigenvector will be;

Now finding eigenvectors  when  

Hence we get;

and

If we take [] we get;

And;

Therefore eigenvector will be;

Hence we can write here;

When eigenvalues is 1;

eigenvector is;

When eigenvalue is 2;

eigenvector is;

When eigenvalue is 3;

eigenvector is;

__Part 3)__

Given matrix is;

Finding eigenvalue of the given matrix;

Hence determinant will be;

Here all the eigenvalues are same.So eigenvector will be only one.

Now finding eigenvectors  when  

Hence we get;

and

If we take [] then, we get;

Therefore eigenvector will be;