Question

Consider the given matrix. 1 −1 0 5 1 4 0 1 1 Find the eigenvalues. (Enter your answers as a comma-separated list.) λ = Find the eigenvectors. (Enter your answers in order of the corresponding eigenvalues, from smallest to largest by real part, then by imaginary part.)

Answer 1

To find the eigenvalues, determine the determinant of (A - λI) and solve the characteristic equation. The eigenvectors are found by solving (A - λI)x = 0 for each eigenvalue.

Step-by-step explanation:

Finding Eigenvalues and Eigenvectors

To find the eigenvalues of a given matrix, we must solve the characteristic equation which is determined by the determinant of (A - λI), where A is the matrix and I is the identity matrix of the same size. The eigenvalues (λ) are the roots of this equation. Once the eigenvalues are found, we can calculate the eigenvectors by solving the equation (A - λI)x = 0 for each eigenvalue, where x represents the eigenvector associated with each eigenvalue.

The given matrix is:

1 -1 0
5 1 4
0 1 1

We will find the eigenvalues by calculating the determinant of (A - λI), and then find the corresponding eigenvectors by solving the system of linear equations for each eigenvalue.

Answer 2

Step-by-step explanation:

We have to diagonalize the matrix

`\left[\begin{array}{ccc}1&-1&0\\5&1&4\\0&1&1\end{array}\right]`

we have to solve the expression

`|A-\lambda I|=0`

Thus, by applying the determinant we obtain the polynomial

`(1-\lambda )^3+5-4=0\\(1-\lambda )^3+1=0\\`

`-\lambda^3+3\lambda^2-4\lambda+2`

`\lambda_1=1\\\lambda_2=1-i\\\lambda_3=1+i\\`

and the eigenvector will be

`v_1=(-4,0,5)\\v_2=(-1,-i,-1)\\v_3=(-1,i,1)`

HOPE THIS HELPS!!