Diagonalize the matrix matA={[2 0 0],[1 4 -1],[-2 -4 4]}

To diagonalize a matrix, we need to find its eigenvalues and eigenvectors.

Step 1: Find the eigenvalues The eigenvalues are the roots of the characteristic equation, which is given by det(A - λI) = 0, where A is the given matrix, λ are the eigenvalues, I is the identity matrix and det denotes the determinant.

For the given matrix A = {[2 0 0],[1 4 -1],[-2 -4 4]}, the characteristic equation is:

det(A - λI) = det({[2-λ 0 0],[1 4-λ -1],[-2 -4 4-λ]}) = 0

This simplifies to (2-λ)((4-λ)(4-λ) - ((-1)*(-2))) = 0

Solving this equation gives the eigenvalues λ1 = 2, λ2 = 4, λ3 = 4.

Step 2: Find the eigenvectors The eigenvectors are found by plugging each eigenvalue back into the equation (A - λI)v = 0 and solving for v.

For λ1 = 2, we get the system of equations:

(2-2)v1 + 0v2 + 0v3 = 0 v1 + (4-2)v2 - v3 = 0 -2v1 - 4v2 + (4-2)v3 = 0

This simplifies to:

0 = 0 v1 + 2v2 - v3 = 0 -2v1 - 4v2 + 2v3 = 0

From the second equation, we can express v1 in terms of v2 and v3: v1 = v3 - 2v2. Substituting this into the third equation gives -2v3 + 4v2 + 2v3 - 4v2 = 0, which is always true. Therefore, the eigenvector corresponding to λ1 = 2 is v1 = (1, 0, 1).

Repeating this process for λ2 = λ3 = 4 gives the eigenvectors v2 = (0, 1, -2) and v3 = (0, 0, 0).

Step 3: Diagonalize the matrix The diagonalized matrix D is formed by placing the eigenvalues along the diagonal, and the matrix P is formed by placing the corresponding eigenvectors as columns. Therefore, the diagonalized form of A is given by PDP^-1, where P^-1 is the inverse of P.

In this case, D = {[2 0 0],[0 4 0],[0 0 4]} and P = {[1 0 0],[0 1 0],[1 -2 0]}. Therefore, the diagonalized form of A is given by PDP^-1 = {[2 0 0],[1 4 -1],[-2 -4 4]}.