Question

Appreciate if you could show the solution clearly step by step for

me to learn. tqvm
2. Determine all the eigenvalues of the matrix G given below. Hence,obtain an eigenvec- tor corresponding to the largest eigenvalue of matrix G. 2 4 0 G= 4 8 0 00

Answer 1

To determine the eigenvalues of matrix G, subtract the identity matrix scaled by λ from G and set the determinant equal to zero. The eigenvalues of G are λ = 0, λ = 2, and λ = 8. To find an eigenvector corresponding to the largest eigenvalue (λ = 8), substitute λ = 8 back into the equation and solve the system of equations.

Step-by-step explanation:

To determine the eigenvalues of a matrix, we need to find the values of λ that satisfy the equation (G - λI)x = 0, where G is the matrix, λ is the eigenvalue, I is the identity matrix, and x is the eigenvector.

Given the matrix G = [[2, 4, 0], [4, 8, 0], [0, 0, 0]], we subtract the identity matrix scaled by λ from G to get G - λI = [[2-λ, 4, 0], [4, 8-λ, 0], [0, 0, -λ]].

To obtain the eigenvalues, we set the determinant of G - λI equal to zero and solve for λ. The determinant equation is (2-λ)(8-λ)(-λ) = 0.

Solving this equation, we find three eigenvalues: λ = 0, λ = 2, and λ = 8.

Next, to obtain an eigenvector corresponding to the largest eigenvalue (λ = 8), we substitute λ = 8 back into the equation (G - λI)x = 0. This gives us the system of equations 2x + 4y = 0 and 4x + (8-8)y = 0. Solving this system, we find that x = 2y.

Thus, the eigenvector corresponding to the largest eigenvalue is x = 2y, where x and y are the scalar components of the eigenvector.

Answer 2

To find the eigenvalues of matrix G, we create the characteristic equation by subtracting λ times the identity matrix from G and calculating the determinant. The solutions to this equation are the eigenvalues. An eigenvector for the largest eigenvalue can be found by substituting this value into the equation (G - λI)x = 0 and solving for vector x.

Step-by-step explanation:

To determine the eigenvalues of matrix G, we set up the characteristic equation of G, which is |G - λI| = 0, where λ represents an eigenvalue and I is the identity matrix.

The matrix G given is:

G =
2 4 0
4 8 0
0 0 0

To find the eigenvalues, we subtract λ times the identity matrix from G and take the determinant:

|G - λI| =
|2-λ 4 0|
|4 8-λ 0|
|0 0 -λ|

This simplifies to:

(2-λ)(8-λ)(-λ) - (4)(4)(-λ) = 0

Solving this equation will give us the eigenvalues. Once we have the eigenvalues, we find the eigenvectors by solving (G - λI)x = 0 for each eigenvalue λ. To find an eigenvector for the largest eigenvalue, we substitute the largest eigenvalue into the equation and solve for the vector x, which will be the eigenvector.11